# To find the value of g ″ ( π 6 ) .

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 5.4, Problem 22E
To determine

## To find the value of g″(π6) .

Expert Solution

The value of g(π6) is 52 .

### Explanation of Solution

Given information:

The function are f(x)=0sinx1+t2dt and g(y)=3yf(x)dx

Concept used:

The fundamental theorem of calculus, part 1 is defined by,

If f is continuous on [a,b] , then the function g defined by,

g(x)=axf(t)dt

axb is an antiderivative of f , that is g(x)=f(x) for a<x<b .

Calculation:

Using Part 1 of the Fundamental Theorem of Calculus, the derivative of the function is obtained as:

f(x)=0sinx1+t2dt

By, evaluation theorem,

ddx0sinx1+t2dt=F(sinx)F(x)=f(x)=1+sin2x

Now, for g(y)=3yf(x)dx ,

g(y)=3y0sinx1+t2dtdx

Using Part 1 of the Fundamental Theorem of Calculus,

g(y)=3yF(x)dx [since, 1+t2 is continuous and 0<x<b ]

Again, Using Part 1 of the Fundamental Theorem of Calculus,

g(y)=F(y)g(y)=1+sin2yg(π6)=1+sin2(π6)g(π6)=1+(12)2g(π6)=1+14g(π6)=54g(π6)=52

Therefore,

The value of g(π6) is 52 .

### Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!