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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

Example 1

Brooke is strolling through the forest. Her velocity East is given by the following graph:

(a) At what time(s) is Brooke not moving?

(b) On what time interval(s) is Brooke moving East?

(c) On what time interval(s) is Brooke moving West?

(d) At what time(s) is Brooke moving most rapidly East?

(e) At what time(s) is Brooke moving most rapidly West?

(f) At what time(s) is Brooke moving most rapidly?

(g) On what time interval(s) is Brooke speeding up?

(h) On what time interval(s) is Brooke slowing down?

Example 2

A cat starts at ground level and climbs a tree with velocity given by the graph below. When v(t) > 0 the cat is climbing upwards.

How high is the cat when t = 4? When t = 7?

Example 3

A squirrel starts two feet above ground and climbs a tree with velocity given by the graph below. When v(t) > 0 the squirrel is climbing upwards.

How high is the squirrel when t = 7?

Example 4

A bug is crawling around on a number line. The bug's velocity, in units per second, is given by the graph below. Positive velocities indicate that the bug is traveling to the right.

Assume the bug is at 0 on the number line when t = 0 seconds.

(a) Where on the number line is the bug when t = 4,6,8, and 10?

(b) At what time(s) does the bug change direction?

(c) How many times during the interval 0 ≤ t ≤ 10 is the bug at position -8 on the number line?

Example 5

Mara went on an hour-long bicycle ride. She started 15 miles from home and her velocity for the hour is given by the graph below.

When velocity is positive, it means her distance from home is increasing.

How far from home was Mara after

(a) 20 minutes?

(b) 50 minutes?

(c) one hour?

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