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The Fundamental Theorem of Calculus

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)

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