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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

Example 1

When v(t) = -100 feet per second, the cheetah is running South at a speed of 100 feet per second. When v(t) = 10, what direction is the cheetah running?

Example 2

If Jen's velocity is -60 mph, how fast is she going?

Example 3

If a garden snail's velocity is 0 feet per second, how fast is the snail moving?

Example 4

Let v(t) be the velocity of a bug moving on the x-axis, measured in units per second. What does it mean to say v(t) = -4?

Example 5

A hummingbird is flying back and forth with velocity given by the graph below, where positive velocities mean the hummingbird is flying North and negative velocities mean the hummingbird is flying South:

(a) At what time(s) is the hummingbird moving neither North nor South?

(b) On what time interval(s) is the hummingbird moving North?

(c) On what time interval(s) is the hummingbird moving South?

(d) At what time(s) is the hummingbird moving most rapidly North?

(e) At what time(s) is the hummingbird moving most rapidly South?

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

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

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

Example 6

A bug crawling on a number line has velocity given by the graph below, where positive velocities indicate the bug is crawling to the right:

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

(b) On what time interval(s) is the bug moving to the right?

(c) On what time interval(s) is the bug moving to the left?

(d) At what time(s) is the bug moving most rapidly to the right?

(e) At what time(s) is the bug moving most rapidly to the left?

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

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

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

Example 7

A cat climbs up and down a tree with velocity given by the graph below. When the velocity is positive it means the cat is climbing up the tree.

(a) At what time(s) does the cat change direction?

(b) At what time(s) is the cat climbing most rapidly upwards?

(c) At what time(s) is the cat climbing most rapidly downwards?

(d) What is the cat's fastest speed on the interval [0,7]?

(e) What is the cat's slowest speed on the interval [0,7]?

(f) At what time(s) is the cat highest in the tree?

Example 8

A bug crawls back and forth on a number line with velocity given by the graph below. Positive velocities correspond to movement in the positive direction on the number line.

(a) If the bug is at 0 on the number line when t = 0, determine the position of the bug at t = 3, 5, and 8 seconds.

(b) If the bug is at -10 on the number line when t = 0, determine the position of the bug at t = 3, 5, and 8 seconds.

Example 9

The graph below describes the velocity of a car over a 10-hour scenic drive. Positive velocity indicates the car is traveling East.

(a) Select the correct answer: From time t = 0 hours to t = 4 hours the car is traveling (East|West).

(b) Fill in the blanks: From time t = ? to time t = ? the car is traveling West.

(c) At what time(s) is the car stopped?

(d) After 10 hours of driving, is the car East or West of where it started? How far East or West of where it started?

Example 10

A whiny toddler is in the exact center of a 20-foot long room. At one end of the room is a lollipop and at the other end is a teddy bear. The toddler toddles back and forth with velocity given by the graph below. When velocity is positive, it means the toddler is moving towards the teddy bear.

(a) Describe the toddler's location at t = 2, 4, 6, and 10 seconds.

(b) How many times between t = 0 and t = 10 seconds will the toddler pass through the exact center of the room?

(c) Will the toddler eventually reach the lollipop, the teddy bear, or neither?

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