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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.118c
Chapter 4, Problem 4.R.118c

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.




c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.

Verified step by step guidance
1
To find the derivative ƒ'(x) of the function ƒ(x) = (a + x)^x, we will use logarithmic differentiation. Start by taking the natural logarithm of both sides: ln(ƒ(x)) = ln((a + x)^x).
Apply the logarithmic identity ln((a + x)^x) = x * ln(a + x). This simplifies the expression to ln(ƒ(x)) = x * ln(a + x).
Differentiate both sides with respect to x. For the left side, use the chain rule: d/dx[ln(ƒ(x))] = ƒ'(x)/ƒ(x). For the right side, use the product rule: d/dx[x * ln(a + x)] = 1 * ln(a + x) + x * (1/(a + x)).
Combine the derivatives: ƒ'(x)/ƒ(x) = ln(a + x) + x/(a + x). Solve for ƒ'(x) by multiplying both sides by ƒ(x): ƒ'(x) = ƒ(x) * (ln(a + x) + x/(a + x)).
Substitute ƒ(x) = (a + x)^x back into the expression for ƒ'(x): ƒ'(x) = (a + x)^x * (ln(a + x) + x/(a + x)). Now, you can graph ƒ(x) and ƒ'(x) for the given values of a: 0.5, 1, 2, and 3.

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Key Concepts

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

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. For the function ƒ(x) = (a + x)ˣ, applying the rules of differentiation, particularly the product and chain rules, is essential to compute ƒ'.
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Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. In the given function ƒ(x) = (a + x)ˣ, the base of the exponent is not constant, which introduces complexity in its behavior. Understanding the properties of exponential growth and how they affect the graph is crucial for analyzing ƒ and its derivative.
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Graphing Functions

Graphing functions involves plotting the values of a function and its derivative on a coordinate plane to visualize their behavior. For the function ƒ(x) and its derivative ƒ', it is important to consider how changes in the parameter 'a' affect the shape and position of the graphs. This visual representation aids in understanding the relationship between the function and its rate of change.
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Related Practice
Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

e. On what intervals (approximately) is f concave up?

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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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Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

d. Give the approximate coordinates of the zero(s) of f.

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Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

f. On what intervals (approximately) is f concave down?  

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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.


x¹⸍² and x¹⸍³

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