Chapter 4: Q. 6 (page 364)

Find a function f that has the given derivative f and value f(c), if possible.
$f' (x) = 3 \sin (- 2 x) - 5 \cos (3 x), f (0) = 0$

Short Answer

Expert verified

The function f is $f (x) = \frac{3}{2} \cos (- 2 x) - \frac{5}{3} \sin (3 x) + c$ and the value of c is . $\frac{3}{2}$

Step by step solution

Step 1. Given Information.

The derivative:
$f' (x) = 3 \sin (- 2 x) - 5 \cos (3 x), f (0) = 0$

Step 2. Find the function by integration.

Integrating both sides with respect to x,

$\int f' (x) . d x = \int 3 \sin (- 2 x) - 5 \cos (3 x) d x f (x) = \int \sin (- 2 x) d x - 5 \int \cos (3 x) d x = 3 (\frac{1}{2} \cos (- 2 x) - 5 (\frac{1}{3} \sin (3 x) + c = \frac{3}{2} \cos (- 2 x) - \frac{5}{3} \sin (3 x) + c$

where c is a constant.

Step 3. Find the value of c.

Substitute the values in the function,

$f (x) = \frac{3}{2} \cos (- 2 x) - \frac{5}{3} \sin (3 x) + c f (0) = \frac{3}{2} \cos (- 2 (0)) - \frac{5}{3} \sin (3 (0)) + c 0 = \frac{3}{2} (1) - \frac{5}{3} (0) + c c = \frac{3}{2}$

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Find a function f that has the given derivative f and value f(c), if possible.
$f' (x) = 3 \sin (- 2 x) - 5 \cos (3 x), f (0) = 0$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the function by integration.

Step 3. Find the value of c.

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Find a function f that has the given derivative f and value f(c), if possible.f'(x)=3sin(-2x)-5cos(3x),f(0)=0

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the function by integration.

Step 3. Find the value of c.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Logic and Functions

Mechanics Maths

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Find a function f that has the given derivative f and value f(c), if possible.
$f' (x) = 3 \sin (- 2 x) - 5 \cos (3 x), f (0) = 0$