# The Fundamental Theorem of Calculus

### Example 1

Let *s*(*t*) be a position function with velocity function *v*(*t*) = *s*'(*t*).

(a) If *s*(9) = 4 and , what is *s*(12)?

(b) If *s*(3) = -2 and , what is *s*(5)?

### Example 2

A koala is climbing up and down its tree with velocity *v*(*t*) feet per minute, where positive values of *v*(*t*) indicate the koala is climbing up the tree.

At t = 0 minutes the koala is 5 feet above ground.

(a) If feet, how high is the koala after *t* = 4 minutes?

(b) If feet, how high is the koala at *t* = 7 minutes (use part (a))?

### Example 3

A hummingbird flies away from its feeder with velocity *v*(*t*) feet per second and position *s*(*t*) feet away from its feeder.

(a) If *s*(4) = 5 feet and feet, find the hummingbird's distance from the feeder at time *t* = 0.

(b) If *s*(10) = 16 feet and feet, find *s*(5).

### Example 4

Let *s*(*t*) be a position function with velocity function *v*(*t*) = *s*'(*t*).

(a) If and *s*(9) = -3, what is *s*(8)?

(b) If and *s*(7) = -4, what is *s*(3)?

(c) Given (a) and (b), what is ?

(d) Given (a) and (b), what is ?

### Example 5

Let *s*(*t*) be a position function with velocity function *v*(*t*) = *s*'(*t*).

(a) If *s*(2) = 7 and *s*(9) = 13, what is ?

(b) If *s*(2) = 13 and *s*(9) = 7, what is ?

(c) If *s*(2) = 13 and *s*(9) = 7 then what is ? (hint: one of the properties of integrals says how to change the limits of integration)