Question

part A)The antiderivative of an acceleration function is a position function.

A)True

B)False

Part B)A right Riemann sum always gives an underestimate.

A) True

B) False

Part C)Integration by substitution is the process that reverses the chain rule.

A) True

B) False

Part D) The derivative of an indefinite integral returns the original function.

A) True

B) False

Part E) The antiderivative of a constant is linear.

A) True

B) False

Part F) Suppose you want to estimate the area under a function f ( x ) from x = 2 to x = 6 using a Riemann sum and n = 2 rectangles. What is Δ x ?

A) 2

B) 1

C) 4

D) 3

Part G) To compute

∫ ( 2 x + 5 ) e x 2 + 5 x d x

what substitution would you need to make?

A) u=2x+5

B) u=x2+5x

C) u=x

D) None of these answers are correct.

Answer 1

Part (a):

The antiderivative of an acceleration function is a position function.

It is False.

Because, the antiderivative of an acceleration function is a velocity function.

Part (b):

A right Riemann sum always gives an underestimate.

It is False.

Because, Left Riemann sum always gives an underestimate and Right Riemann sum always gives an overestimate.

Part (c):

Integration by substitution is the process that reverses the chain rule.

It is true.

Part (d):

The derivative of an indefinite integral returns the original function.

It is true.

Part (e):

The antiderivative of a constant is linear.

It is true.

It has degree 0 in integral and antiderivative becomes a degree 1.

Part (F):

Given:

x = 2 (a = 2) to x = 6 (b = 6) and n = 2 rectangles.

So, .

Part (g):

Given:

Apply u- substitution,

.

Then integral becomes,

Now substitute u back.

.