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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 4.4.20
Chapter 4, Problem 4.4.20

Graphing functions Use the guidelines of this section to make a complete graph of f.


f(x) = x⁴ + 4x³

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Step 1: Identify the domain of the function f(x) = x⁴ + 4x³. Since this is a polynomial function, the domain is all real numbers.
Step 2: Find the critical points by taking the derivative of f(x). Compute f'(x) = \(\frac{d}{dx}\)(x⁴ + 4x³) using the power rule.
Step 3: Set the derivative f'(x) equal to zero to find the critical points. Solve the equation f'(x) = 0 for x.
Step 4: Determine the behavior of the function at the critical points by using the second derivative test. Compute f''(x) and evaluate it at the critical points to determine concavity.
Step 5: Analyze the end behavior of the function by considering the leading term of f(x). As x approaches infinity or negative infinity, the term x⁴ dominates, indicating the function's behavior at the extremes.

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Key Concepts

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

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, f(x) = x⁴ + 4x³ is a polynomial of degree 4, which indicates its highest exponent. Understanding the general shape and behavior of polynomial functions is crucial for graphing them effectively.
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Introduction to Polynomial Functions

Critical Points and Extrema

Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. To find these points for f(x), we need to compute its derivative, f'(x), and solve for x. Analyzing these points helps in determining the overall shape and turning behavior of the graph.
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Critical Points

End Behavior of Functions

End behavior describes how a function behaves as the input values approach positive or negative infinity. For polynomial functions, the leading term (the term with the highest degree) primarily dictates this behavior. In the case of f(x) = x⁴ + 4x³, as x approaches ±∞, the function will also approach ±∞, which is essential for sketching the graph accurately.
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Related Practice
Textbook Question

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


p(t) = 2t³ + 3t² - 36t

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Textbook Question

Solving initial value problems Find the solution of the following initial value problems.

g'(x) = 7x(x⁶ - 1/7); g(1) = 2

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Textbook Question

Explain how to apply the First Derivative Test.

Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ ((1 + √x)/x)dx

Textbook Question

{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = cos x

Textbook Question

Suppose the position of an object moving horizontally after seconds is given by the function s(t) = 32t - t⁴, where 0 ≤ t ≤ 3 and s is measured in feet, with s > 0 corresponding to positions to the right of the origin. When is the object farthest to the right?