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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 3.10.65b
Chapter 3, Problem 3.10.65b

62–65. {Use of Tech} Graphing f and f'
b. Compute and graph f'.
f(x)=e^−x tan^−1 x on [0,∞)

Verified step by step guidance
1
Step 1: Understand the function f(x) = e^(-x) * tan^(-1)(x). This function is a product of two functions: e^(-x) and tan^(-1)(x). To find the derivative f'(x), we will use the product rule.
Step 2: Recall the product rule for derivatives, which states that if you have two functions u(x) and v(x), then the derivative of their product is given by (uv)' = u'v + uv'. Here, let u(x) = e^(-x) and v(x) = tan^(-1)(x).
Step 3: Compute the derivative of u(x) = e^(-x). The derivative u'(x) is found using the chain rule: u'(x) = -e^(-x).
Step 4: Compute the derivative of v(x) = tan^(-1)(x). The derivative v'(x) is 1/(1 + x^2).
Step 5: Apply the product rule to find f'(x): f'(x) = u'(x)v(x) + u(x)v'(x) = (-e^(-x))tan^(-1)(x) + e^(-x)/(1 + x^2). Now, use a graphing tool to plot both f(x) and f'(x) over the interval [0, ∞).

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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 changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For the function f(x) = e^(-x) * arctan(x), the derivative f'(x) can be computed using the product rule and chain rule of differentiation.
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Derivatives

Graphing Functions

Graphing a function involves plotting its output values against its input values on a coordinate system. For the function f(x) = e^(-x) * arctan(x), understanding its behavior as x approaches 0 and infinity is crucial. The graph of f' will provide insights into the function's increasing or decreasing behavior, as well as its critical points.
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Behavior at Infinity

Analyzing the behavior of a function as x approaches infinity helps determine its long-term trends. For f(x) = e^(-x) * arctan(x), as x increases, e^(-x) approaches 0, while arctan(x) approaches π/2. This interplay affects the overall behavior of f and its derivative f', which is essential for understanding the function's limits and asymptotic behavior.
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Related Practice
Textbook Question

{Use of Tech} Bungee jumper A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e^−t cos t), for t ≥ 0.

b. Use a graphing utility to determine when she is moving downward and when she is moving upward during the first 10 s.  

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Textbook Question

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

limx→2(x2−3)5−1x−2{\(\displaystyle\]\lim\)_{x\(\to\)2}}\(\frac{\left(x^2-3\right)^5-1}{x-2}\)

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Textbook Question

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(s) = 4s³+3s; a= -3, -1

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Textbook Question

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.

b. When does the stone reach its highest point?

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Textbook Question

Product Rule for three functions Assume f, g, and h are differentiable at x.

b. Use the formula in (a) to find d/dx(e^x(x−1)(x+3))

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Textbook Question

Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>


b. What is the rate of change of θ with respect to the bird’s height when it is 60 ft above the ground?

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