Module 18 - The Fundamental Theorem

Module 18 - Answers

Lesson 1

Answer 1

18.1.1 The area under the curve between x = 0 and x = 5 is

Answer 2

18.1.2 The area under the curve f(x) = x² between x = 0 and x = b is given by

Answer 3

18.1.3

The area function associated with f(x) = x³ is .

Answer 4

18.1.4 The function for the area under f(x) = x⁴ between 0 and x is

Answer 5

18.1.5

Answer 6

18.1.6 The derivative of the area function A is the curve function f.

Lesson 2

Answer 1

18.2.1

Answer 2

18.2.2

Notice that the coefficient of lnx is approximately 1 and the constant is approximately 0, as denoted by scientific notation E^-4.

Answer 3

18.2.3

Answer 4

18.2.4
The derivative of the area function

is the curve function, or in other words

Answer 5

18.2.5

The area function increases at an increasing rate. It is concave upward.

Lesson 3

Answer 1

18.3.1 The derivative of this integral function is sin(x)/x.

Answer 2

18.3.2 If

, then

and

. By the chain rule,

. Thus,

[-2, 2, 1] x [-5, 5, 1]

Answer 3

18.3.3

Answer 4

18.3.4

Answer 5

18.3.5 -cos(x) is an antiderivative of sin(x), so the integral equals -cos(

) - (-cos(0))=2.

Answer 6

18.3.6

-cos(x²) - (-cos(x))= -cosx² + cosx

Self Test

Answer 1

The area function is

Answer 2

The derivative of the area function with respect to x is 3x² which is f(x).

Answer 3

The graph of the area function has the shape shown below.

Answer 4

Answer 5

sin(x²) - sin(x)