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Chapter 4: Q. 4 (page 364)

Find a function f that has the given derivative f' and value f(c), if possible.
$f' (x) = 3 - {\frac{4}{5}}^{x}, f (1) = 0$

Short Answer

Expert verified

The function is $3 x - \frac{{\frac{4}{5}}^{x}}{\log {\frac{4}{5}}^{}} - 2.992$ .

The value of f(2.992( is $2.992$

Step by step solution

01

Step 1. Given

The given derivative of function is $f' (x) = 3 - {\frac{4}{5}}^{x}, f (1) = 0$

02

Step 2. Finding the function using integration

$f' (x) = 3 - {\frac{4}{5}}^{x}, f (1) = 0 鈭� f' (x) = 鈭� 3 - {\frac{4}{5}}^{x} d x = 3 x - \frac{{\frac{4}{5}}^{x}}{\log {\frac{4}{5}}^{}} + c where c is constant f (1) = 3 (1) - \frac{{\frac{4}{5}}^{1}}{\log {\frac{4}{5}}^{}} + c 0 = 3 - \frac{4}{5} 脳 0.0969 + c 0 = 2.992 + c c = - 2.992 Hence the function is 3 x - \frac{{\frac{4}{5}}^{x}}{\log {\frac{4}{5}}^{}} - 2.992$

03

Step 3. Finding the value of f(c).

$f (x) = 3 x - \frac{{\frac{4}{5}}^{x}}{\log {\frac{4}{5}}^{}} - 2.992 P u t t i n g v a l u e o f c = 2.992 f (c) = 3 脳 2.992 \frac{{\frac{4}{5}}^{2.992}}{\log {\frac{4}{5}}^{2.992}} - 2.992 f (c) = 3.45$

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Most popular questions from this chapter

Explain why we call the collection of antiderivatives of a function f a family. How are the antiderivatives of a function related?

If ${鈭�}_{- 2}^{3} f (x) d x = 4, {鈭�}_{- 2}^{6} f (x) d x = 9, {鈭�}_{- 2}^{3} g (x) d x = 2$ and ${鈭�}_{3}^{6} g (x) d x = 3$ ,then find the values of each definite integral in Exercises $29 - 40$ . If there is not enough information, explain why.

${鈭�}_{3}^{6} f (x) d x$ .

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

$鈭� \frac{3}{x^{2} + 1} d x$

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{鈭� 3}^{2} 鈥� \frac{2 x}{x^{2} + 5} d x$

Without calculating any sums or definite integrals, determine the values of the described quantities. (Hint: Sketch graphs first.)

(a) The signed area between the graph of f(x) = cos x and the x-axis on [鈭捪€, 蟺].

(b) The average value of f(x) = cos x on [0, 2蟺].

(c) The area of the region between the graphs of f(x) = $\sqrt{4 鈭� x^{2}} and g (x) = 鈭� \sqrt{4 鈭� x^{2}} on [鈭� 2, 2] .$

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