Calc Midterm 3

Question 1

Net Change Theorem (Concept)

Answer 1

The integral of the rate of change (derivative) is the total net change

Question 2

Net Change Equation

Answer 2

∫b ->a F’(x)dx = F(b)-F(a)

Question 3

To find the end amount of the Net Change Theorem

Answer 3

F(b) =F(a) + ∫b ->a F’(x)dx

Question 4

If f(x) is the slope of a trail at a distance of x miles what does ∫5->3 f(x)dx represent?

Answer 4

The total change of elevation from x=3 -> 5

Question 5

A honey bee population starts with 100 bees and increases at a rate of n’(t) bees per week. What does 100 + ∫15->0 n’(t)dt represent?

Answer 5

The total number of bees in the population after 15 weeks

Question 6

Process of Net Change Calculation

Answer 6

1. Label your rate, your x, and your total number (pick a letter, just not O)
2. Write an expression for one small piece of that quantity (dC = d(x)dx) then plug in
3. Write the definite integral
4. Evaluate the integral
5. Write a sentence to answer the question

Question 7

Accumulation Function

Answer 7

F(x) = F(a) + ∫x -> a F’(t) dt

Question 8

Review 6. (c) of CW #14

Answer 8

Make sure that equation is set up right

Question 9

MVT

Answer 9

∫b->a f(x)dx = f(x*)(b-a)

Notice how its the opposite of finding the average value. f(x*) IS f(avg)

Question 10

Average Value

Answer 10

f(avg) = 1/(b-a)∫b-a f(x) dx

Question 11

Area Between Curves Equation

Answer 11

∫b->a [f(x) - g(x)] dx

Question 12

Don’t mix up

Answer 12

radius and diameter!

Question 13

Area of an equilateral triangle

Answer 13

(sqrt(3))/4*s^2

Question 14

Volume equation (cross section)

Answer 14

dV = A(x) * h

Question 15

For volume, you equation is in terms of (x, y)

Answer 15

Whatever its perpendicular to

Question 16

Volume equation (disks and washers)

Answer 16

πr^2h or π[(R)^2-(r)^2]

Question 17

Arc length

Answer 17

L = ∫b->a sqrt(1 + (f’(x))^2) dx

Question 18

Another equation for arc length

Answer 18

dL = sqrt((dx/dt)^2 + (dy/dt)^2) dt

Question 19

Surface Area Equation

Answer 19

∫b->a 2π f(x) sqrt(1+[F’(x)]^2) dx

Question 20

Volume Equation: Cylindrical Shells

Answer 20

2pi[1/2(r2+r1)]h(r2-r1)