The document is a lecture note on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions like arcsin, arccos, arctan and gives their domains, ranges and other properties. It also provides examples of calculating the values of inverse trig functions like arcsin(1/2) = π/6 and arctan(-1) = -π/4.
Lesson 13: Exponential and Logarithmic Functions (handout)Matthew Leingang
This document contains lecture notes from a Calculus I class covering exponential and logarithmic functions. The notes discuss definitions and properties of exponential functions, including the number e and the natural exponential function. Conventions for rational, irrational and negative exponents are also defined. Examples are provided to illustrate approximating exponential expressions with irrational exponents using rational exponents. The objectives are to understand exponential functions, their properties, and apply laws of logarithms including the change of base formula.
Lesson 12: Linear Approximations and Differentials (handout)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...Matthew Leingang
This document is a lecture on derivatives of exponential and logarithmic functions from a Calculus I class at New York University. It covers the objectives and outline, which include finding derivatives of exponential functions with any base, logarithmic functions with any base, and using logarithmic differentiation. It provides proofs and examples of finding derivatives, such as the derivative of the natural exponential function being itself and the derivative of the natural logarithm function.
Lesson 20: Derivatives and the Shapes of Curves (handout)Matthew Leingang
This document contains lecture notes on calculus from a Calculus I course. It covers determining the monotonicity of functions using the first derivative test. Key points include using the sign of the derivative to determine if a function is increasing or decreasing over an interval, and using the first derivative test to classify critical points as local maxima, minima, or neither. Examples are provided to demonstrate finding intervals of monotonicity for various functions and applying the first derivative test.
Lesson 13: Exponential and Logarithmic Functions (Section 021 handout)Matthew Leingang
This document contains lecture notes from a Calculus I class at New York University on October 21, 2010 covering sections 3.1-3.2 on exponential functions. The notes include announcements about an upcoming midterm exam and homework assignment. Statistics on the recent midterm exam are provided, showing the average, median, standard deviation, and what constitutes a "good" or "great" score. The objectives of sections 3.1-3.2 are outlined as understanding exponential functions, their properties, and laws of logarithms. The notes provide definitions and derivations of exponential functions for various exponent values.
This paper analyzes the variations in joint angles of a two link planar manipulator used for welding applications. The inverse kinematics equations are used to calculate the joint angles (θ1 and θ2) of the manipulator for a given position of the end effector moving linearly between points A and B. A program is written in Fortran 90 to calculate the joint angles and link positions at multiple step points along the welding seam, as solving the inverse kinematics equations by hand for each point would be complicated. The program uses homogeneous transformations and geometric relationships based on the Denavit-Hartenberger representation to determine the manipulator configuration corresponding to each end effector position.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
The document is a lecture on derivatives of exponential and logarithmic functions. It begins with announcements about homework and an upcoming midterm. It then provides objectives and an outline for sections on exponential and logarithmic functions. The body of the document defines exponential functions, establishes conventions for exponents of all types, discusses properties of exponential functions, and graphs various exponential functions. It focuses on setting up the necessary foundations before discussing derivatives of these functions.
What is Odd about Binary Parseval FramesMicah Bullock
This document examines the construction and properties of binary Parseval frames. It addresses when a binary Parseval frame has a complementary Parseval frame, and which binary symmetric idempotent matrices are Gram matrices of binary Parseval frames. Unlike real or complex Parseval frames, binary Parseval frames do not always have complements. A necessary condition for a binary Parseval frame to have a complement is that it contains at least one even vector. Certain symmetric idempotent matrices that are not Gram matrices of binary Parseval frames can exist if they only have even column vectors.
Reading the Lindley-Smith 1973 paper on linear Bayes estimatorsChristian Robert
The document outlines a seminar on Bayes estimates for the linear model. It introduces the linear model and Bayesian methods. It then discusses exchangeability, providing an example of an exchangeable distribution. It also discusses the general Bayesian linear model, including the posterior distribution of the parameters using a three stage model.
This document discusses using linear approximations to estimate functions. It provides an example estimating sin(61°) using linear approximations about a=0 and a=60°. When approximating about a=0, the estimate is 1.06465. When approximating about a=60°, the estimate is 0.87475, which is closer to the actual value of sin(61°) according to a calculator check. The document teaches that the tangent line provides the best linear approximation near a point, and its equation can be used to estimate function values.
This document discusses character tables and their application in group theory. It contains the following key points:
1. Character tables describe the transformation properties of molecular coordinates and basis functions under different symmetry operations.
2. Group theory can be used to predict chirality, polarity, hybrid orbitals, molecular orbitals, and vibrational spectra of molecules based on their symmetry properties.
3. Molecular vibrations are classified as infrared (IR) active if the vibration changes the dipole moment and Raman active if it changes the polarizability of the molecule, based on the symmetry of the dipole moment and polarizability.
This document presents a new iterative method called the Parametric Method of Iteration for solving nonlinear systems of equations. The method rewrites each equation using a set of positive parameters and iterates the solutions until convergence within a desired accuracy. The method converges faster than traditional iteration and Newton-Raphson methods, as shown through examples. The parametric method generalizes the existing methods and allows tuning of parameters to accelerate convergence for different types of equations.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
The document outlines the syllabus for Class IX mathematics for the academic session 2011-2012. It is divided into two semesters. The first semester covers chapters on number systems, polynomials, coordinate geometry, introduction to Euclid's geometry, lines and angles, triangles, and Heron's formula. The second semester covers chapters on linear equations in two variables, quadrilaterals, areas of parallelograms and triangles, circles, constructions, statistics, probability, and surface areas and volumes. Mental maths practice is scheduled every Monday based on the concerned topic. Related activities are provided at the end of each chapter.
The document discusses weighted nuclear norm minimization and its applications to image denoising. It provides background on key concepts from linear algebra and optimization theory needed to understand the denoising problem, such as convex optimization, affine transformations, singular value decomposition, and eigendecomposition. The objective of denoising is to extract the low-rank original image from a noisy high-dimensional image, modeled as the sum of the original image and white noise.
This document discusses variational principles and Lagrange's equations. It covers Hamilton's principle, the calculus of variations, deriving Lagrange's equations from Hamilton's principle, Hamilton's principle for nonholonomic systems, and conservation theorems. Key points include using Hamilton's principle to find the path that makes the action integral stationary, using the calculus of variations to find such paths, deriving Lagrange's equations by making the variation of the action integral equal to zero, and handling constraints using Lagrange multipliers.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It covers objectives like differentiating constant, sum, and difference functions. It also reviews derivatives of sine and cosine. Examples are provided, like finding the derivative of a squaring function x^2 using the definition of a derivative. The document outlines the topics and provides explanations and step-by-step solutions.
This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Lesson 27: Integration by Substitution (Section 041 slides)Mel Anthony Pepito
The document provides notes from a Calculus I class at New York University on December 13, 2010. It includes announcements about upcoming review sessions and the final exam. It then outlines the objectives and topics to be covered, which is integration by substitution. Examples are provided to demonstrate how to use u-substitution to evaluate indefinite integrals. The key steps are to let u equal an expression involving x, find du in terms of dx, and then substitute into the integral.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few more good examples.
This lecture covers various solution methods for unconstrained optimization problems, including:
1) Line search methods like dichotomous search and the Fibonacci and golden section methods for one-dimensional problems.
2) Newton's method and the false position method for curve fitting to minimize functions in one dimension.
3) Descent methods for multidimensional problems like the gradient method, which follows the negative gradient direction at each step, and how scaling can improve convergence rates.
The document discusses optimizing the area of a rectangular field using 320 yards of fencing. It poses the problem of determining the shape of the field that maximizes the enclosed area given the fixed amount of fencing. It also provides a link to an online quiz about derivatives and graphing that contains problems requiring deep thought about concepts learned.
This document is from a Calculus I class at New York University and covers continuity. It provides announcements about office hours and homework deadlines. It then discusses the objectives of understanding the definition of continuity and applying it to piecewise functions. Examples are provided to demonstrate how to show a function is continuous at a point by evaluating the limit as x approaches the point and showing it equals the function value. The students are asked to determine at which other points the example function is continuous.
This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
Presentation deck used at the Model Schools Conference in Orlando 2012. Presentation on KP Compass and how we use game theory to increase student engagement in our concept driven mastery system. www.kpcompass.com
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.
This document provides an overview of calculating limits from a Calculus I course at New York University. It begins with announcements about homework being due and includes sections on recalling the concept of limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The document uses examples, definitions, proofs, and the error-tolerance game to explain how to calculate limits, limit laws, and applying algebra to limits. It provides the essential information and objectives for understanding how to compute elementary limits.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few more good examples.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...Mel Anthony Pepito
This document is from a Calculus I class at New York University and covers derivatives of exponential and logarithmic functions. It includes objectives, an outline, explanations of properties and graphs of exponential and logarithmic functions, and derivations of derivatives. Key points covered are the derivatives of exponential functions with any base equal the function times a constant, the derivative of the natural logarithm function, and using logarithmic differentiation to find derivatives of more complex expressions.
This document discusses optimization of functions with multiple variables subject to equality constraints. It introduces the method of constrained variation and the method of Lagrange multipliers to find the extremum of a function subject to one or more equality constraints. For a specific example with two variables and one constraint, it derives the necessary conditions using both methods. It then generalizes the necessary and sufficient conditions to problems with n variables and m equality constraints, defining the Lagrangian and determining the equations that must be satisfied at an extremum.
- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals.
- It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.
Methods from Mathematical Data Mining (Supported by Optimization)SSA KPI
This document summarizes a presentation on cluster stability estimation and determining the optimal number of clusters in a dataset. The presentation proposes a method that draws random samples from the dataset and compares the partitions obtained from each sample to estimate cluster stability. It quantifies the consistency between partitions using minimal spanning trees and the Friedman-Rafsky test statistic. Experiments on synthetic and real-world datasets show that the method can accurately determine the true number of clusters by finding the partition that maximizes cluster stability.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
This document contains lecture notes on inverse trigonometric functions. It begins with definitions of inverse functions and conditions for a function to have an inverse. It then defines the inverse trigonometric functions arcsin, arccos, arctan, and arcsec and gives their domains and ranges. Examples are provided to illustrate calculating values of the inverse trigonometric functions. The document concludes with a brief discussion of notational ambiguity and an outline mentioning that derivatives of the inverse trigonometric functions will be covered.
This document contains lecture notes on inverse trigonometric functions. It begins with definitions of inverse functions and conditions for a function to have an inverse. It then defines the inverse trigonometric functions arcsin, arccos, arctan, and arcsec and gives their domains and ranges. Examples are provided to illustrate calculating values of the inverse trigonometric functions. The document concludes with a brief discussion of notational ambiguity and an outline mentioning derivatives of inverse trigonometric functions and applications.
This document is from a Calculus I class at New York University and covers inverse trigonometric functions. It begins with announcements about midterm grades and an upcoming quiz. The objectives are to learn the definitions, domains, ranges and derivatives of inverse trig functions such as arcsin, arccos, arctan, arcsec and arccsc. Examples are provided to demonstrate calculating values of these inverse functions.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
This document contains notes on inverse trigonometric functions including:
- Definitions and graphs of arcsin, arccos, arctan, and arcsec functions
- Derivations of the derivatives of arcsin, arccos, and arctan using the Inverse Function Theorem
- Examples of composing inverse trig functions and finding their derivatives
This document contains notes on inverse trigonometric functions including:
- Definitions and graphs of arcsin, arccos, arctan, and arcsec functions
- Derivations of the derivatives of arcsin, arccos, and arctan using the Inverse Function Theorem
- Examples of composing inverse trig functions and finding their derivatives
We go over the trigonometric function, their inverses, and the derivatives of the inverse functions. The surprising fact is that these derivatives are simpler functions than the functions themselves.
We go over the trigonometric function, their inverses, and the derivatives of the inverse functions. The surprising fact is that these derivatives are simpler functions than the functions themselves.
The document outlines topics to be covered in Calculus I class sessions on the derivative and rates of change, including: defining the derivative at a point and using it to find the slope of the tangent line to a curve at that point; examples of derivatives modeling rates of change; and how to find the derivative function and second derivative of a given function. It provides learning objectives, an outline of topics, and an example problem worked out graphically and numerically to illustrate finding the slope of the tangent line.
The document outlines topics to be covered in Sections 2.1-2.2 of a Calculus I course, including defining the derivative, finding derivatives of functions, relating the graph of a function to its derivative, and finding tangent lines. It provides examples of how derivatives can model real-world rates of change and outlines the objectives to understand the definition of the derivative and how to apply it to functions.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...Mel Anthony Pepito
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
This document provides an outline and learning objectives for a midterm exam covering vectors and three-dimensional coordinate systems in a Math 21a course. The midterm will cover material up to and including section 11.4 in the textbook. It outlines key topics like three-dimensional coordinate systems, vectors, the dot and cross product, equations of lines and planes, and vector functions. Examples are provided for distance between points in space and rewriting an equation in standard form to identify what surface it represents. Learning objectives are stated for topics like three-dimensional coordinate systems, vectors, and vector addition.
This document is from a Calculus I course at New York University. It outlines the objectives and content for Sections 2.1-2.2 on the derivative and rates of change. The sections will define the derivative, discuss how it models rates of change, and how to find the derivative of various functions graphically and numerically. It will also cover how to find higher-order derivatives and sketch the derivative graph from a function graph.
This document is a lecture summary for Calculus I at New York University. It covers the objectives and outline for sections 2.1 and 2.2 on the derivative and rates of change. The summary uses examples and graphics to demonstrate how to find the slope of the tangent line to a curve at a given point, both graphically and numerically using the definition of the derivative.
The document is a lecture note from an NYU Calculus I class covering the definite integral. It provides announcements about upcoming quizzes and exams. The content discusses computing definite integrals using Riemann sums and the limit of Riemann sums, as well as properties of the definite integral. Examples are given of calculating Riemann sums using different representative points in each interval.
This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.
The document discusses the definite integral, including computing it using Riemann sums, estimating it using approximations like the midpoint rule, and reasoning about its properties. It outlines the topics to be covered, such as recalling previous concepts and comparing properties of integrals. Formulas are provided for calculating Riemann sums using different representative points within the intervals.
Similar to Lesson 16: Inverse Trigonometric Functions (Section 041 slides) (20)
This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.
This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMel Anthony Pepito
The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of
The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.
The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.
This document outlines information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences. It provides details on the course staff, contact information for the professor, an overview of assessments including homework, quizzes, a midterm and final exam. Grading breakdown is also included, as well as information on purchasing the required textbook and accessing the course on Blackboard. The document aims to provide students with essential logistical information to succeed in the Calculus I course.
This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It includes announcements about homework and deadlines. It then discusses guidelines for written homework assignments and a rubric for grading. The bulk of the document explains the concept of limit intuitively through an "error-tolerance game" and provides examples to illustrate the idea. It aims to build an informal understanding of limits before providing a precise definition.
No, this is not a function because the same input of 2 is mapped to two different outputs of 4 and 5. For a relation to be a function, each input must map to a unique output.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
This document is a section from a Calculus I course at New York University dated September 20, 2010. It discusses continuity of functions, beginning with definitions and examples of determining continuity. Key points covered include the definition of continuity as a function having a limit equal to its value, and theorems stating polynomials, rational functions, and combinations of continuous functions are also continuous. Trigonometric functions like sin, cos, tan, and cot are shown to be continuous on their domains. The document provides examples, explanations, and questions to illustrate the concept of continuity.
This document is from a Calculus I course at New York University and covers limits involving infinity. It discusses definitions of limits approaching positive or negative infinity. Examples are provided of functions with infinite limits, such as 1/x as x approaches 0. The document outlines techniques for finding limits at points where a function is not continuous, such as using a number line to determine the signs of factors in a rational function's denominator.
1. Section 3.5
Inverse Trigonometric
Functions
V63.0121.041, Calculus I
New York University
November 1, 2010
Announcements
Midterm grades have been submitted
Quiz 3 this week in recitation on Section 2.6, 2.8, 3.1, 3.2
Thank you for the evaluations
. . . . . .
2. Announcements
Midterm grades have been
submitted
Quiz 3 this week in
recitation on Section 2.6,
2.8, 3.1, 3.2
Thank you for the
evaluations
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 2 / 32
3. Objectives
Know the definitions,
domains, ranges, and
other properties of the
inverse trignometric
functions: arcsin, arccos,
arctan, arcsec, arccsc,
arccot.
Know the derivatives of the
inverse trignometric
functions.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 3 / 32
4. What is an inverse function?
Definition
Let f be a function with domain D and range E. The inverse of f is the
function f−1 defined by:
f−1 (b) = a,
where a is chosen so that f(a) = b.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 4 / 32
5. What is an inverse function?
Definition
Let f be a function with domain D and range E. The inverse of f is the
function f−1 defined by:
f−1 (b) = a,
where a is chosen so that f(a) = b.
So
f−1 (f(x)) = x, f(f−1 (x)) = x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 4 / 32
6. What functions are invertible?
In order for f−1 to be a function, there must be only one a in D
corresponding to each b in E.
Such a function is called one-to-one
The graph of such a function passes the horizontal line test: any
horizontal line intersects the graph in exactly one point if at all.
If f is continuous, then f−1 is continuous.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 5 / 32
8. arcsin
Arcsin is the inverse of the sine function after restriction to [−π/2, π/2].
y
.
. . . x
.
π π s
. in
−
. .
2 2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 7 / 32
9. arcsin
Arcsin is the inverse of the sine function after restriction to [−π/2, π/2].
y
.
.
. . . x
.
π π s
. in
−
. . .
2 2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 7 / 32
10. arcsin
Arcsin is the inverse of the sine function after restriction to [−π/2, π/2].
y
.
y
. =x
.
. . . x
.
π π s
. in
−
. . .
2 2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 7 / 32
11. arcsin
Arcsin is the inverse of the sine function after restriction to [−π/2, π/2].
y
.
. . rcsin
a
.
. . . x
.
π π s
. in
−
. . .
2 2
.
The domain of arcsin is [−1, 1]
[ π π]
The range of arcsin is − ,
2 2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 7 / 32
12. arccos
Arccos is the inverse of the cosine function after restriction to [0, π]
y
.
c
. os
. . x
.
0
. .
π
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 8 / 32
13. arccos
Arccos is the inverse of the cosine function after restriction to [0, π]
y
.
.
c
. os
. . x
.
0
. .
π
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 8 / 32
14. arccos
Arccos is the inverse of the cosine function after restriction to [0, π]
y
.
y
. =x
.
c
. os
. . x
.
0
. .
π
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 8 / 32
15. arccos
Arccos is the inverse of the cosine function after restriction to [0, π]
. . rccos
a
y
.
.
c
. os
. . . x
.
0
. .
π
.
The domain of arccos is [−1, 1]
The range of arccos is [0, π]
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 8 / 32
16. arctan
Arctan is the inverse of the tangent function after restriction to
[−π/2, π/2]. y
.
. x
.
3π π π 3π
−
. −
. . .
2 2 2 2
t
.an
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 9 / 32
17. arctan
Arctan is the inverse of the tangent function after restriction to
[−π/2, π/2]. y
.
. x
.
3π π π 3π
−
. −
. . .
2 2 2 2
t
.an
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 9 / 32
18. arctan
y
. =x
Arctan is the inverse of the tangent function after restriction to
[−π/2, π/2]. y
.
. x
.
3π π π 3π
−
. −
. . .
2 2 2 2
t
.an
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 9 / 32
19. arctan
Arctan is the inverse of the tangent function after restriction to
[−π/2, π/2]. y
.
π
. a
. rctan
2
. x
.
π
−
.
2
The domain of arctan is (−∞, ∞)
( π π)
The range of arctan is − ,
2 2
π π
lim arctan x = , lim arctan x = −
x→∞ 2 x→−∞ 2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 9 / 32
20. arcsec
Arcsecant is the inverse of secant after restriction to
[0, π/2) ∪ (π, 3π/2]. y
.
. x
.
3π π π 3π
−
. −
. . .
2 2 2 2
s
. ec
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 10 / 32
21. arcsec
Arcsecant is the inverse of secant after restriction to
[0, π/2) ∪ (π, 3π/2]. y
.
.
. x
.
3π π π 3π
−
. −
. . . .
2 2 2 2
s
. ec
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 10 / 32
22. arcsec
Arcsecant is the inverse of secant after restriction to . = x
y
[0, π/2) ∪ (π, 3π/2]. y
.
.
. x
.
3π π π 3π
−
. −
. . . .
2 2 2 2
s
. ec
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 10 / 32
23. arcsec 3π
.
2
Arcsecant is the inverse of secant after restriction to
[0, π/2) ∪ (π, 3π/2]. . . y
π
.
2 .
. . x
.
.
The domain of arcsec is (−∞, −1] ∪ [1, ∞)
[ π ) (π ]
The range of arcsec is 0, ∪ ,π
2 2
π 3π
lim arcsec x = , lim arcsec x =
x→∞ 2 x→−∞ 2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 10 / 32
24. Values of Trigonometric Functions
π π π π
x 0
6 4 3 2
√ √
1 2 3
sin x 0 1
2 2 2
√ √
3 2 1
cos x 1 0
2 2 2
1 √
tan x 0 √ 1 3 undef
3
√ 1
cot x undef 3 1 √ 0
3
2 2
sec x 1 √ √ 2 undef
3 2
2 2
csc x undef 2 √ √ 1
2 3
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 11 / 32
25. Check: Values of inverse trigonometric functions
Example
Find
arcsin(1/2)
arctan(−1)
( √ )
2
arccos −
2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 12 / 32
27. What is arctan(−1)?
.
3
. π/4
.
. .
.
−
. π/4
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 13 / 32
28. What is arctan(−1)?
.
( )
3
. π/4 3π
. Yes, tan = −1
4
√
2
s
. in(3π/4) =
2
.
√ .
2
. os(3π/4) = −
c
2
.
−
. π/4
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 13 / 32
29. What is arctan(−1)?
.
) (
3
. π/4 3π
. Yes, tan = −1
4
√ But, the)
( π π range of arctan is
2
s
. in(3π/4) = − ,
2 2 2
.
√ .
2
. os(3π/4) = −
c
2
.
−
. π/4
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 13 / 32
30. What is arctan(−1)?
.
(
)
3
. π/4 3π
. Yes, tan = −1
4
But, the)
( π π range of arctan is
√ − ,
2 2 2
c
. os(π/4) =
. 2 Another angle whose
. π
tangent is −1 is − , and
√ 4
2 this is in the right range.
. in(π/4) = −
s
2
.
−
. π/4
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 13 / 32
31. What is arctan(−1)?
.
(
)
3
. π/4 3π
. Yes, tan = −1
4
But, the)
( π π range of arctan is
√ − ,
2 2 2
c
. os(π/4) =
. 2 Another angle whose
. π
tangent is −1 is − , and
√ 4
2 this is in the right range.
. in(π/4) = −
s π
2 So arctan(−1) = −
4
.
−
. π/4
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 13 / 32
34. Caution: Notational ambiguity
. in2 x =.(sin x)2
s . in−1 x = (sin x)−1
s
sinn x means the nth power of sin x, except when n = −1!
The book uses sin−1 x for the inverse of sin x, and never for
(sin x)−1 .
1
I use csc x for and arcsin x for the inverse of sin x.
sin x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 15 / 32
36. The Inverse Function Theorem
Theorem (The Inverse Function Theorem)
Let f be differentiable at a, and f′ (a) ̸= 0. Then f−1 is defined in an
open interval containing b = f(a), and
1
(f−1 )′ (b) = ′ −1
f (f (b))
In Leibniz notation we have
dx 1
=
dy dy/dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 17 / 32
37. The Inverse Function Theorem
Theorem (The Inverse Function Theorem)
Let f be differentiable at a, and f′ (a) ̸= 0. Then f−1 is defined in an
open interval containing b = f(a), and
1
(f−1 )′ (b) = ′ −1
f (f (b))
In Leibniz notation we have
dx 1
=
dy dy/dx
Upshot: Many times the derivative of f−1 (x) can be found by implicit
differentiation and the derivative of f:
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 17 / 32
38. Illustrating the Inverse Function Theorem
.
Example
Use the inverse function theorem to find the derivative of the square root
function.
. . . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 18 / 32
39. Illustrating the Inverse Function Theorem
.
Example
Use the inverse function theorem to find the derivative of the square root
function.
Solution (Newtonian notation)
√
Let f(x) = x2 so that f−1 (y) = y. Then f′ (u) = 2u so for any b > 0 we have
1
(f−1 )′ (b) = √
2 b
. . . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 18 / 32
40. Illustrating the Inverse Function Theorem
.
Example
Use the inverse function theorem to find the derivative of the square root
function.
Solution (Newtonian notation)
√
Let f(x) = x2 so that f−1 (y) = y. Then f′ (u) = 2u so for any b > 0 we have
1
(f−1 )′ (b) = √
2 b
Solution (Leibniz notation)
If the original function is y = x2 , then the inverse function is defined by x = y2 .
Differentiate implicitly:
dy dy 1
1 = 2y =⇒ = √
dx dx 2 x
. . . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 18 / 32
41. Derivation: The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
cos y = 1 =⇒ = =
dx dx cos y cos(arcsin x)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 19 / 32
42. Derivation: The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
cos y = 1 =⇒ = =
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 19 / 32
43. Derivation: The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
cos y = 1 =⇒ = =
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
1
.
x
.
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 19 / 32
44. Derivation: The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
cos y = 1 =⇒ = =
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
1
.
x
.
y
. = arcsin x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 19 / 32
45. Derivation: The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
cos y = 1 =⇒ = =
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
1
.
x
.
y
. = arcsin x
. √
. 1 − x2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 19 / 32
46. Derivation: The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
cos y = 1 =⇒ = =
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
√
cos(arcsin x) = 1 − x2 1
.
x
.
y
. = arcsin x
. √
. 1 − x2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 19 / 32
47. Derivation: The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
cos y = 1 =⇒ = =
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
√
cos(arcsin x) = 1 − x2 1
.
x
.
So
d 1
arcsin(x) = √ y
. = arcsin x
dx 1 − x2 . √
. 1 − x2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 19 / 32
48. Graphing arcsin and its derivative
1
.√
1 − x2
The domain of f is [−1, 1],
but the domain of f′ is . . rcsin
a
(−1, 1)
lim f′ (x) = +∞
x→1−
lim f′ (x) = +∞ .
| . .
|
x→−1+ −
. 1 1
.
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 20 / 32
49. Composing with arcsin
Example
Let f(x) = arcsin(x3 + 1). Find f′ (x).
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 21 / 32
50. Composing with arcsin
Example
Let f(x) = arcsin(x3 + 1). Find f′ (x).
Solution
We have
d 1 d 3
arcsin(x3 + 1) = √ (x + 1)
dx 1 − (x3 + 1)2 dx
3x2
=√
−x6 − 2x3
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 21 / 32
51. Derivation: The derivative of arccos
Let y = arccos x, so x = cos y. Then
dy dy 1 1
− sin y = 1 =⇒ = =
dx dx − sin y − sin(arccos x)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 22 / 32
52. Derivation: The derivative of arccos
Let y = arccos x, so x = cos y. Then
dy dy 1 1
− sin y = 1 =⇒ = =
dx dx − sin y − sin(arccos x)
To simplify, look at a right
triangle:
√
sin(arccos x) = 1 − x2 1
. √
. 1 − x2
So
d 1 y
. = arccos x
arccos(x) = − √ .
dx 1 − x2 x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 22 / 32
53. Graphing arcsin and arccos
. . rccos
a
. . rcsin
a
.
| . |.
.
−
. 1 1
.
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 23 / 32
54. Graphing arcsin and arccos
. . rccos
a
Note
(π )
cos θ = sin −θ
. . rcsin
a 2
π
=⇒ arccos x = − arcsin x
2
.
| . |.
. So it’s not a surprise that their
−
. 1 1
. derivatives are opposites.
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 23 / 32
55. Derivation: The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = 1 =⇒ = = cos2 (arctan x)
dx dx sec2 y
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 24 / 32
56. Derivation: The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = 1 =⇒ = = cos2 (arctan x)
dx dx sec2 y
To simplify, look at a right
triangle:
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 24 / 32
57. Derivation: The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = 1 =⇒ = = cos2 (arctan x)
dx dx sec2 y
To simplify, look at a right
triangle:
x
.
.
1
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 24 / 32
58. Derivation: The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = 1 =⇒ = = cos2 (arctan x)
dx dx sec2 y
To simplify, look at a right
triangle:
x
.
y
. = arctan x
.
1
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 24 / 32
59. Derivation: The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = 1 =⇒ = = cos2 (arctan x)
dx dx sec2 y
To simplify, look at a right
triangle:
√
. 1 + x2 x
.
y
. = arctan x
.
1
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 24 / 32
60. Derivation: The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = 1 =⇒ = = cos2 (arctan x)
dx dx sec2 y
To simplify, look at a right
triangle:
1
cos(arctan x) = √
1 + x2 √
. 1 + x2 x
.
y
. = arctan x
.
1
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 24 / 32
61. Derivation: The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = 1 =⇒ = = cos2 (arctan x)
dx dx sec2 y
To simplify, look at a right
triangle:
1
cos(arctan x) = √
1 + x2 √
. 1 + x2 x
.
So
d 1 y
. = arctan x
arctan(x) = .
dx 1 + x2
1
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 24 / 32
62. Graphing arctan and its derivative
y
.
. /2
π
a
. rctan
1
.
1 + x2
. x
.
−
. π/2
The domain of f and f′ are both (−∞, ∞)
Because of the horizontal asymptotes, lim f′ (x) = 0
x→±∞
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 25 / 32
63. Composing with arctan
Example
√
Let f(x) = arctan x. Find f′ (x).
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 26 / 32
64. Composing with arctan
Example
√
Let f(x) = arctan x. Find f′ (x).
Solution
d √ 1 d√ 1 1
arctan x = (√ )2 x= · √
dx 1+ x dx 1+x 2 x
1
= √ √
2 x + 2x x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 26 / 32
65. Derivation: The derivative of arcsec
Try this first.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 27 / 32
66. Derivation: The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 27 / 32
67. Derivation: The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 27 / 32
68. Derivation: The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 27 / 32
69. Derivation: The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
x
.
.
1
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 27 / 32
70. Derivation: The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
x
.
y
. = arcsec x
.
1
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 27 / 32
71. Derivation: The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
√
x2 − 1
tan(arcsec x) = √
1
x
. . x2 − 1
y
. = arcsec x
.
1
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 27 / 32
72. Derivation: The derivative of arcsec
Try this first. Let y = arcsec x, so x = sec y. Then
dy dy 1 1
sec y tan y = 1 =⇒ = =
dx dx sec y tan y x tan(arcsec(x))
To simplify, look at a right
triangle:
√
x2 − 1
tan(arcsec x) = √
1
x
. . x2 − 1
So
d 1
arcsec(x) = √ .
y
. = arcsec x
dx x x2 − 1
1
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 27 / 32
73. Another Example
Example
Let f(x) = earcsec 3x . Find f′ (x).
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 28 / 32
74. Another Example
Example
Let f(x) = earcsec 3x . Find f′ (x).
Solution
1
f′ (x) = earcsec 3x · √ ·3
3x (3x)2 − 1
3earcsec 3x
= √
3x 9x2 − 1
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 28 / 32
76. Application
Example
One of the guiding principles of
most sports is to “keep your
eye on the ball.” In baseball, a
batter stands 2 ft away from
home plate as a pitch is thrown
with a velocity of 130 ft/sec
(about 90 mph). At what rate
does the batter’s angle of gaze
need to change to follow the
ball as it crosses home plate?
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 30 / 32
77. Application
Example
One of the guiding principles of
most sports is to “keep your
eye on the ball.” In baseball, a
batter stands 2 ft away from
home plate as a pitch is thrown
with a velocity of 130 ft/sec
(about 90 mph). At what rate
does the batter’s angle of gaze
need to change to follow the
ball as it crosses home plate?
Solution
Let y(t) be the distance from the ball to home plate, and θ the angle the
batter’s eyes make with home plate while following the ball. We know
y′ = −130 and we want θ′ at the moment that y = 0.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 30 / 32
78. Solution
Let y(t) be the distance from the ball to home plate, and θ the angle the
batter’s eyes make with home plate while following the ball. We know
y′ = −130 and we want θ′ at the moment that y = 0.
y
.
1
. 30 ft/sec
.
θ
.
. 2
. ft
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 31 / 32
79. Solution
Let y(t) be the distance from the ball to home plate, and θ the angle the
batter’s eyes make with home plate while following the ball. We know
y′ = −130 and we want θ′ at the moment that y = 0.
We have θ = arctan(y/2). Thus
dθ 1 1 dy
= ·
dt 1 + (y/2)2 2 dt
y
.
1
. 30 ft/sec
.
θ
.
. 2
. ft
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 31 / 32
80. Solution
Let y(t) be the distance from the ball to home plate, and θ the angle the
batter’s eyes make with home plate while following the ball. We know
y′ = −130 and we want θ′ at the moment that y = 0.
We have θ = arctan(y/2). Thus
dθ 1 1 dy
= ·
dt 1 + (y/2)2 2 dt
When y = 0 and y′ = −130,
then y
.
dθ 1 1
= · (−130) = −65 rad/sec 1
. 30 ft/sec
dt y=0 1+0 2
.
θ
.
. 2
. ft
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 31 / 32
81. Solution
Let y(t) be the distance from the ball to home plate, and θ the angle the
batter’s eyes make with home plate while following the ball. We know
y′ = −130 and we want θ′ at the moment that y = 0.
We have θ = arctan(y/2). Thus
dθ 1 1 dy
= ·
dt 1 + (y/2)2 2 dt
When y = 0 and y′ = −130,
then y
.
dθ 1 1
= · (−130) = −65 rad/sec 1
. 30 ft/sec
dt y=0 1+0 2
The human eye can only track .
θ
.
at 3 rad/sec! . 2
. ft
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 31 / 32
82. Summary
y y′
1
arcsin x √
1 − x2
1
arccos x − √ Remarkable that the
1 − x2
derivatives of these
1
arctan x transcendental functions
1 + x2 are algebraic (or even
1 rational!)
arccot x −
1 + x2
1
arcsec x √
x x2 − 1
1
arccsc x − √
x x2 − 1
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 32 / 32