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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.R.34
Chapter 4, Problem 4.R.34

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.


{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ

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Identify the domain of the function. Since the function is a product of polynomials and an exponential function, it is defined for all real numbers.
Find the first derivative, f'(x), to determine the critical points and intervals of increase and decrease. Use the product rule and chain rule: f'(x) = d/dx [x(x-1)e^(-x)].
Set the first derivative equal to zero to find critical points: f'(x) = 0. Solve for x to find where the function changes from increasing to decreasing or vice versa.
Find the second derivative, f''(x), to determine concavity and points of inflection. Use the product rule and chain rule again: f''(x) = d/dx [f'(x)].
Analyze the behavior of the function as x approaches positive and negative infinity to determine end behavior. Consider the limits of f(x) as x approaches infinity and negative infinity.

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Key Concepts

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

Curve Sketching

Curve sketching involves analyzing a function's behavior to create a visual representation of its graph. This includes determining key features such as intercepts, asymptotes, intervals of increase and decrease, and concavity. By applying calculus concepts like derivatives, one can identify critical points and inflection points, which are essential for accurately sketching the curve.
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Summary of Curve Sketching

Derivatives

Derivatives represent the rate of change of a function and are fundamental in understanding its behavior. By finding the first derivative, one can determine where the function is increasing or decreasing, while the second derivative helps identify concavity and points of inflection. These insights are crucial for sketching the graph accurately and understanding the function's overall shape.
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Derivatives

Exponential Functions

Exponential functions, such as e^(-x), exhibit unique properties, including rapid growth or decay. In the given function f(x) = x(x - 1)e^(-x), the exponential component influences the function's behavior as x approaches infinity or negative infinity. Understanding how exponential functions interact with polynomial terms is essential for predicting the function's end behavior and overall graph shape.
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Related Practice
Textbook Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = sin 2x + 3 on [-π , π]

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Textbook Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = 4x¹⸍² - x⁵⸍² on [0, 4]

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Textbook Question

{Use of Tech} Newton’s method Use Newton’s method to approximate the roots of ƒ(x) = e⁻²ˣ + 2eˣ - 6 to six digits.

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Textbook Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→∞ ln ((x +1) / (x-1))

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Textbook Question

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.


2ˣ and 4ˣ⸍²

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Textbook Question

Optimization A right triangle has legs of length h and r and a hypotenuse of length 4 (see figure). It is revolved about the leg of length h to sweep out a right circular cone. What values of h and r maximize the volume of the cone? (Volume of a cone = (1/3) πr²h.) <IMAGE>

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