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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 5
Chapter 3, Problem 5

Let f(x) = sin x. What is the value of f′(π)?

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First, identify the function f(x) = sin(x). We need to find the derivative of this function, which is f′(x).
Recall the derivative rule for the sine function: the derivative of sin(x) with respect to x is cos(x). Therefore, f′(x) = cos(x).
Now, substitute x = π into the derivative function f′(x) = cos(x) to find f′(π).
Evaluate cos(π). Remember that the cosine of π radians is a known trigonometric value.
Conclude by stating the value of f′(π) based on the evaluation of cos(π).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, we need to find the derivative of the sine function, which will help us determine the slope of the tangent line at a specific point.
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Derivatives

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to side lengths in right triangles. The sine function, specifically, gives the ratio of the length of the opposite side to the hypotenuse in a right triangle. Understanding the properties and values of these functions at key angles (like π) is essential for evaluating derivatives and solving related problems.
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Introduction to Trigonometric Functions

Value of f′(x) at Specific Points

To find the value of the derivative at a specific point, we evaluate the derivative function at that point. For the sine function, we first compute its derivative, which is cosine. Then, we substitute the specific angle (in this case, π) into the derivative to find the slope of the tangent line at that point, which provides the answer to the original question.
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Initial Value Problems
Related Practice
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.

Textbook Question

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>


c. [0,1][0, 1]

Textbook Question

Derivatives using tables Let h(x)=f(g(x))h(x)=f(g(x)) and p(x)=g(f(x))p(x)=g(f(x)). Use the table to compute the following derivatives.

<IMAGE>

e. h′(5)h^{\(\prime\)}\(\left\)(5\(\right\))

1
views
Textbook Question

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>


a. [0,2][0, 2]

Textbook Question

Derivatives using tables Let h(x)=f(g(x))h(x)=f(g(x)) and p(x)=g(f(x))p(x)=g(f(x)). Use the table to compute the following derivatives.

<IMAGE>

d. p′(2)p^{\(\prime\)}\(\left\)(2\(\right\))

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Textbook Question

A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water.

b. At what rate is the volume of the water increasing if the water level is rising at 1/4ft/min.