Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.9.5

Chapter 4, Problem 4.9.5

Give the antiderivatives of xᵖ . For what values of p does your answer apply?

Verified step by step guidance

Step 1: Recall the general formula for the antiderivative of a power function xᵖ. The antiderivative of xᵖ is given by: ∫xᵖ dx = (1/(p+1)) * x^(p+1) + C, where C is the constant of integration.

Step 2: Identify the condition under which this formula is valid. The formula applies when p ≠ -1, because dividing by (p+1) would result in division by zero if p = -1.

Step 3: For the special case where p = -1, the antiderivative of x⁻¹ is ∫x⁻¹ dx = ∫(1/x) dx = ln|x| + C. This is derived from the natural logarithm function.

Step 4: Summarize the result. The antiderivative of xᵖ is (1/(p+1)) * x^(p+1) + C for p ≠ -1, and ln|x| + C for p = -1.

Step 5: Ensure clarity by emphasizing that the constant of integration, C, must always be included in the final expression for the antiderivative.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Antiderivative

An antiderivative of a function is another function whose derivative is the original function. In calculus, finding the antiderivative is a fundamental operation, often referred to as integration. For example, the antiderivative of x^p is (x^(p+1))/(p+1), provided p is not equal to -1.

Power Rule for Integration

The Power Rule for Integration states that the integral of x raised to a power p is given by (x^(p+1))/(p+1) + C, where C is the constant of integration. This rule applies when p is not equal to -1, as the case p = -1 leads to the natural logarithm function, ln|x|, instead.

Domain of the Antiderivative

The domain of the antiderivative refers to the values of p for which the antiderivative formula is valid. Specifically, the formula (x^(p+1))/(p+1) applies for all real numbers p except p = -1, where the function x^p becomes x^(-1) and its antiderivative is ln|x|, indicating a different behavior at that point.

Recommended video:

05:50

Antiderivatives

Related Practice

Textbook Question

Explain the Mean Value Theorem with a sketch.

Textbook Question

Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.

f'(x) > 0, for all x in the domain of f'; f'(-2) and f'(1) do not exist; f"(0) = 0

Textbook Question

Minimum distance Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)?

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.

f(x) = 2x³ - 3x² + 12

Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0⁺ x²ˣ

Textbook Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = x² √(x + 5)

views