Book cover for Calculus

Calculus

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

ISBN #9781119379331

7th Edition

5,101 Questions

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33,688 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section focuses on constructing antiderivatives both graphically and numerically. By analyzing graphs of derivative functions, one can sketch the corresponding antiderivatives through initial conditions and vertical shifts. The Fundamental Theorem of Calculus provides the formal basis for relating definite integrals to changes in antiderivative values. Moreover, the concept that all antiderivatives of a function differ by a constant underlines the significance of the constant of integration in expressing general solutions.

Learning Objectives

1

Explain how to construct antiderivatives graphically using slope functions and initial conditions.

2

Apply the Fundamental Theorem of Calculus to compute function values from derivative graphs.

3

Demonstrate the impact of different initial conditions through vertical shifts of antiderivative graphs.

4

Understand the general form of an antiderivative and the role of the constant of integration.

Key Concepts

CONCEPT

DEFINITION

Antiderivative

A function F(x) whose derivative is f(x), i.e., F'(x) = f(x).

Fundamental Theorem of Calculus (FTC)

A theorem that relates the integral of a derivative f'(x) over [a, b] to the difference F(b) - F(a).

Initial Condition

A given value of the antiderivative, such as F(a) = C, used to determine a specific antiderivative from the family of solutions.

Constant of Integration

An arbitrary constant added to the antiderivative, reflecting that any two antiderivatives of the same function differ by a constant.

Indefinite Integral

The notation ∫ f(x) dx, which represents the general antiderivative F(x) + C of the function f(x).

Example Problems

Example 1

Fill in the blanks in the following statements, assuming that $F(x)$ is an antiderivative of $f(x)$ (a) If $f(x)$ is positive over an interval, then $F(x)$ is over the interval. (b) If $f(x)$ is increasing over an interval, then $F(x)$ is over the interval.

Example 2

Use Figure 6.10 and the fact that $P=0$ when $t=0$ to find values of $P$ when $t=1,2,3,4$ and 5 (FIGURE CAN'T COPY)

Example 3

Use Figure 6.11 and the fact that $P=2$ when $t=0$ to find values of $P$ when $t=1,2,3,4$ and 5 (FIGURE CAN'T COPY)

Example 4

Let $G^{\prime}(t)=g(t)$ and $G(0)=4 .$ Use Figure 6.12 to find the values of $G(t)$ at $t=5,10,20,25$ (FIGURE CAN'T COPY)

Example 5

Sketch two functions $F$ such that $F^{\prime}=f$ In one case let $F(0)=0$ and in the other, let $F(0)=1$. (FIGURE CAN'T COPY)

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Step-by-Step Explanations

QUESTION

How can you sketch several antiderivative curves given a derivative’s graph and different initial conditions?

STEP-BY-STEP ANSWER:

Step 1: Identify the slope provided by the derivative graph at each x-value to determine where the function is increasing or decreasing.
Step 2: Begin at the initial condition point on the vertical axis (e.g., F(0)=0, F(0)=1, or F(0)=2).
Step 3: Draw the function by following the slopes indicated by the derivative, ensuring that regions with positive and negative slopes are correctly represented.
Step 4: Recognize that differing initial conditions result in vertical shifts; the curves have the same shape but different starting points.
Final Answer: The antiderivative graphs are obtained by vertically shifting a single curve, consistent with the slopes indicated by the derivative.

Sketching Antiderivatives Graphically

QUESTION

How do you compute the value of an antiderivative at a point using a known initial condition and a definite integral?

STEP-BY-STEP ANSWER:

Step 1: Start with the known initial condition, F(a).
Step 2: Write the relation F(b) = F(a) + ∫[a to b] f'(x) dx as provided by the Fundamental Theorem of Calculus.
Step 3: Evaluate the definite integral using the area under the derivative’s graph, taking into account positive areas above the x-axis and negative areas below.
Step 4: Add the calculated integral (net area) to the initial condition to obtain F(b).
Final Answer: F(b) is obtained by adding the net area under f'(x) from a to b to F(a), i.e., F(b) = F(a) + ∫[a to b] f'(x) dx.

Computing Antiderivatives Using Definite Integrals

QUESTION

Why do all antiderivatives of a function differ only by a constant?

STEP-BY-STEP ANSWER:

Step 1: Start with two functions F and G that are both antiderivatives of f, meaning F'(x) = f(x) and G'(x) = f(x).
Step 2: Form the difference H(x) = G(x) - F(x).
Step 3: Differentiate H(x) to obtain H'(x) = G'(x) - F'(x) = f(x) - f(x) = 0.
Step 4: By the Constant Function Theorem, if H'(x) = 0 then H(x) must be a constant.
Final Answer: All antiderivatives differ by a constant because their difference has a zero derivative, hence is constant.

General Antiderivatives and the Constant of Integration

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Common Mistakes

  • Assuming that different antiderivative curves have different slopes rather than being vertical shifts of the same function.
  • Forgetting to include the constant of integration when writing the general antiderivative.
  • Misestimating areas under the curve, especially neglecting that areas below the x-axis subtract from the total.
  • Confusing the derivative’s role in producing instantaneous slopes with the actual function values obtained from integration.