2. Given a position­time function, s = f(t) we differentiate to find
the velocity, v = ds
dt
Given a velocity­time function, we integrate to find the total
b
net distance traveled, s(b) ­ s(a) = s '(t) dt
a

3. change in f = rate of change * time
f(b)
change of f = f(b) ­ f(a)
f(a)
a b

5. Fundamental Theorem of Calculus
Let the function f be continuous on [a, b] with derivative f '.
Then
b
total change in f = f(b) ­ f(a) = f ' (t) dt
a
* If we know f(a), the Fundamental Theorem enables us to
reconstruct the function from a knowledge of its derivative

6. The price of a new car is $24 500. The price of a new car is
changing at a rate of dP = 120 + 180 t dollars per year.
dt
How much will the car cost 5 years from now?

7. Change in price =
5 5
P(5) ­ P(0) = P '(t) dt = ( 120 + 180 t ) dt
0 0
Calculate an RSUM

10. Given the function f(x) = x3 the derivative is:
2
Compute
1
a) left and right sums with 50 subdivisions
b) the Fundamental Theorem of Calculus

12. The velocity of a car in km per hour is given by
v(t) = 3t2 + 2t for t 0
Calculate the distance traveled from t = 1 to t = 4 hours
a) using the left and right sums with n = 100
b) using the Fundamental Theorem and the fact that if
f(t) = t3 + t2 f '(t) = 3t2 + 2t

13. The function f(x) = sin(2x) has the derivative f '(x) = 2 cos(2x)
/2
Compute 2cos(2x) dx using
0
a) left and right sums with 50 subdivisions
b) the Fundamental Theorem of Calculus

14. The graph below shows the rate in gallons per hour at
which oil is leaking out of a tank
y = r(t)

15. Write a definite integral that represents the total amount
of oil that leaks out in the first hour.
y = r(t)
1
r(t) dt
0

16. Shade the region whose area represents the total amount
of oil that leaks out in the first hour.
y = r(t)

17. Give a lower and upper estimate of the total amount of oil
that leaks out in the first hour.
y = r(t)