Textbook Question
Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
d. Verify that the results of parts (a) and (c) are consistent.
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.10.61d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. The lines tangent to the graph of y=sin x on the interval [−π/2,π/2] have a maximum slope of 1.
Verified step by step guidance1
First, recall that the slope of the tangent line to the graph of a function y = f(x) at a point x is given by the derivative f'(x). For the function y = sin(x), the derivative is f'(x) = cos(x).
Next, consider the interval [−π/2, π/2]. We need to find the maximum value of the derivative f'(x) = cos(x) on this interval, as this will give us the maximum slope of the tangent lines.
The function cos(x) is continuous and differentiable on the interval [−π/2, π/2]. To find its maximum value, we can evaluate cos(x) at the critical points and endpoints of the interval.
The critical points occur where the derivative of cos(x), which is -sin(x), is zero. This happens when x = 0 within the interval [−π/2, π/2]. Evaluate cos(x) at x = 0, which gives cos(0) = 1.
Finally, evaluate cos(x) at the endpoints of the interval: cos(−π/2) = 0 and cos(π/2) = 0. Comparing these values, the maximum value of cos(x) on the interval [−π/2, π/2] is indeed 1, confirming that the maximum slope of the tangent lines is 1.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Lines
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point. For the function y = sin x, the slope of the tangent line is given by its derivative, which is cos x.
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Slopes of Tangent Lines
Derivatives
The derivative of a function measures how the function's output value changes as its input value changes. For y = sin x, the derivative is cos x, which indicates the slope of the tangent line at any point x. Understanding the behavior of the derivative within a specific interval helps determine maximum and minimum slopes.
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Maximum and Minimum Values
In calculus, finding the maximum and minimum values of a function involves analyzing its critical points and endpoints within a given interval. For the function y = sin x on the interval [−π/2, π/2], we can evaluate the derivative to find where it reaches its highest value, which is essential for determining the maximum slope of the tangent lines.
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Related Practice
Textbook Question
97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>
{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.
d. Graph P' and use the graph to estimate the year in which the population is growing fastest.
Textbook Question
Tangents and inverses Suppose L(x)=ax+b (with a≠0) is the equation of the line tangent to the graph of a one-to-one function f at (x0,y0). Also, suppose M(x)=cx+d is the equation of the line tangent to the graph of f^−1 at (y0,x0).
c. Prove that L^−1(x)=M(x).
Textbook Question
Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>
d. Determine the velocity of the airliner at noon (t = 6) and explain why the velocity is negative.
Textbook Question
Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>
d. f'(1)
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Textbook Question
{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.
d. At what time is the magnitude of the flow rate a minimum? A maximum?