Q. 12 Suppose f is a function whose de... [FREE SOLUTION]

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Chapter 4: Q. 12 (page 372)

Suppose f is a function whose derivative $f'$ is given by the graph shown next. In Exercises 9鈥�12, use the given value of f , an area approximation, and the Fundamental Theorem to approximate the requested value
Given that $f (- 1) = 2$ , approximate $f (1)$ .

Short Answer

Expert verified

The required approximation $f (1) = 5.17$

Step by step solution

Step 1. Given information

Given that $f (- 1) = 2$ and graph of $f'$

Step 2. Finding the value of f(1)

Use the method. If fis derivative of the function $f'$ then they by fundamental theorem $f (x) = 鈭� f' (x) d x + C$

The graph (parabola) of the function shows that $f'$ is quadratic function whose roots are 2.5 and -0.5

Therefore, $f' (x) = (x - 2.5) (x - 0.5) = x^{2} - 3 x + 1.25 N o w, f (x) = f' (x) d x + C, w h e r e C i s a c o n s \tan t$

$鈬� f (x) = 鈭� (x^{2} - 3 x + 1.25) d x + C 鈬� f (x) = \frac{x^{3}}{3} - 3 \frac{x^{2}}{2} + 1.25 x + C . . . . . . . . . . . . (1)$

we have $f (- 1) = 2 . . . . . . . . . . . (2)$

From results (1) and (2)

$f (- 1) = \frac{{(- 1)}^{3}}{3} - 3 \frac{{(- 1)}^{2}}{2} + 1.25 (- 1) + C 鈬� f (- 1) = - \frac{1}{3} - \frac{3}{2} - \frac{5}{4} + C 鈬� 2 = - \frac{1}{3} - \frac{3}{2} - \frac{5}{4} + C C = 2 + \frac{1}{3} + \frac{3}{2} + \frac{5}{4} = \frac{24 + 4 + 18 + 15}{12} = \frac{61}{12}$

Substitute value of C in result (1)

$f (x) = \frac{x^{3}}{3} - 3 \frac{x^{2}}{2} + \frac{5}{4} x + \frac{61}{12} . . . . . . . . . . . . . . . . . . . . . . . (3)$

$f (1) = \frac{{(1)}^{3}}{3} - 3 \frac{{(1)}^{2}}{2} + \frac{5}{4} (1) + \frac{61}{12} = \frac{1}{3} - \frac{3}{2} + \frac{5}{4} + \frac{61}{12} 鈬� f (1) = \frac{4 - 18 + 15 + 61}{12} = \frac{62}{12} = 5.17$

Thus, $f (1) = 5.17$

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91影视

Suppose f is a function whose derivative $f'$ is given by the graph shown next. In Exercises 9鈥�12, use the given value of f , an area approximation, and the Fundamental Theorem to approximate the requested value
Given that $f (- 1) = 2$ , approximate $f (1)$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the value of f(1)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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91影视

Suppose f is a function whose derivative f' is given by the graph shown next. In Exercises 9鈥�12, use the given value of f , an area approximation, and the Fundamental Theorem to approximate the requested valueGiven that f(-1)=2, approximate f(1).

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the value of f(1)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Probability and Statistics

Discrete Mathematics

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Suppose f is a function whose derivative $f'$ is given by the graph shown next. In Exercises 9鈥�12, use the given value of f , an area approximation, and the Fundamental Theorem to approximate the requested value
Given that $f (- 1) = 2$ , approximate $f (1)$ .