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Chapter 4: Q. 6 (page 364)

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

Expert verified

The function f is f(x)=32cos(-2x)-53sin(3x)+c and the value of c is .32

Step by step solution

01

Step 1. Given Information.

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

02

Step 2. Find the function by integration.

Integrating both sides with respect to x,

鈭�f'(x).dx=鈭�3sin(-2x)-5cos(3x)dxf(x)=鈭�sin(-2x)dx-5鈭�cos(3x)dx=3(12cos(-2x)-5(13sin(3x)+c=32cos(-2x)-53sin(3x)+c

where c is a constant.

03

Step 3. Find the value of c.

Substitute the values in the function,

f(x)=32cos(-2x)-53sin(3x)+cf(0)=32cos(-2(0))-53sin(3(0))+c0=32(1)-53(0)+cc=32

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Most popular questions from this chapter

Explain why at this point we don鈥檛 have an integration formula for the functionf(x)=secx whereas we do have an integration formula for f(x)=sinx.

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�23x+1dx

Prove that 鈭�13(3x+4)dx=20in three different ways:

(a) algebraically, by calculating a limit of Riemann sums;

(b) geometrically, by recognizing the region in question as a trapezoid and calculating its area;

(c) with formulas, by using properties and formulas of definite integrals.

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value.

limn鈫�鈭�鈭�k=1nk2+k+1n

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. (Hint for Exercise 54: tanx=sinxcosx).

鈭�(tanx+xsec2x)dx.
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