Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

2. Intro to Derivatives

Tangent Lines and Derivatives

Calculus

2. Intro to Derivatives

Tangent Lines and Derivatives

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1:55 minutes

Problem 3.10.61d

Textbook Question

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 guidance

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.

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.

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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Textbook Question

Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.

73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.

c. f(x) = √|x-4|

Textbook Question

Graph the following curves and determine the location of any vertical tangent lines.

a. x²+y² = 9

Textbook Question

Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.

ƒ(x) = √(1-2x)

Textbook Question

Find an equation of the line tangent to the following curves at the given value of x.

y = 4 sin x cos x; x = π/3

Textbook Question

The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>

Compute the average rate of population growth from 1950 to 1960.

Textbook Question

Explain why the average rate of growth from 1950 to 1960 is a good approximation to the (instantaneous) rate of growth in 1955.

Textbook Question

Estimate the instantaneous rate of growth in 1985.

Textbook Question

72–76. Tangent lines Find an equation of the line tangent to each of the following curves at the given point.

y = 3x³+ sin x; (0, 0)

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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.

Key Concepts

Tangent Lines

Derivatives

Maximum and Minimum Values

Watch next

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.