Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.10.65c

Chapter 3, Problem 3.10.65c

62–65. {Use of Tech} Graphing f and f'
c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.
f(x)=e^−x tan^−1 x on [0,∞)

Verified step by step guidance

Step 1: Understand the problem. We need to graph the function f(x) = e^(-x) * tan^(-1)(x) and its derivative f'(x) over the interval [0, ∞). We will verify that the zeros of f'(x) correspond to points where f(x) has a horizontal tangent line.

Step 2: Find the derivative f'(x). Use the product rule for differentiation, which states that if you have a function h(x) = u(x) * v(x), then h'(x) = u'(x) * v(x) + u(x) * v'(x). Here, u(x) = e^(-x) and v(x) = tan^(-1)(x).

Step 3: Differentiate u(x) and v(x) separately. The derivative of u(x) = e^(-x) is u'(x) = -e^(-x). The derivative of v(x) = tan^(-1)(x) is v'(x) = 1/(1 + x^2).

Step 4: Apply the product rule. Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula to find f'(x). This gives f'(x) = -e^(-x) * tan^(-1)(x) + e^(-x) * (1/(1 + x^2)).

Step 5: Graph f(x) and f'(x) using a graphing tool. Identify the zeros of f'(x) on the graph. These zeros are the x-values where f'(x) = 0, indicating that f(x) has a horizontal tangent line at these points. Verify that these correspond to the horizontal tangents on the graph of f(x).

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Key Concepts

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

Derivative and Critical Points

The derivative of a function, denoted as f', represents the rate of change of the function f. Critical points occur where the derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection. In this context, finding the zeros of f' helps identify where the function f has horizontal tangent lines, which are essential for understanding the function's behavior.

Horizontal Tangent Lines

A horizontal tangent line occurs at points on the graph of a function where the slope is zero, meaning the derivative f' equals zero. This indicates that the function is neither increasing nor decreasing at that point, which is crucial for identifying local extrema. Verifying that the zeros of f' correspond to horizontal tangents helps confirm the relationship between the derivative and the function's graphical behavior.

Graphing Functions

Graphing a function involves plotting its values on a coordinate system to visualize its behavior. For the function f(x) = e^(-x) tan^(-1)(x), understanding its graph helps in analyzing its critical points and the nature of its tangent lines. Technology can assist in graphing to easily identify where the function has horizontal tangents, enhancing comprehension of the relationship between f and f'.

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Textbook Question

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x)=(x²−1)sin^−1 x on [−1,1]

Textbook Question

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c. Find the average velocity of the car over the interval [1.75, 2.25]. Estimate the velocity of the car at 11:00 A.M. and determine the direction in which the patrol car is moving.

Textbook Question

Deriving trigonometric identities

c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.

Textbook Question

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

c. Suppose the population density of the city remains constant from year to year at 1000 people mi². Determine the growth rate of the population in 2030.

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Textbook Question

Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position $y equals 0$ when the mass hangs at rest. Suppose you push the mass to a position $y sub 0$ units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is $y = y_{0} \cos (t \sqrt{\frac{k}{m}})$ , where $k is greater than 0$ is a constant measuring the stiffness of the spring (the larger the value of $k$ , the stiffer the spring) and $y$ is positive in the upward direction.

Use equation (4) to answer the following questions.

c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness ( $k$ is increased by a factor of $4$ )?

Textbook Question

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>

c. d/dx ((f(x)g(x)) |x=3

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62–65. {Use of Tech} Graphing f and f'c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.f(x)=e^−x tan^−1 x on [0,∞)

Verified video answer for a similar problem:

Key Concepts

Derivative and Critical Points

Horizontal Tangent Lines

Graphing Functions

62–65. {Use of Tech} Graphing f and f'
c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.
f(x)=e^−x tan^−1 x on [0,∞)