# The Fundamental Theorem of Calculus

### Topics

## Introduction to The Fundamental Theorem Of Calculus - At A Glance:

Velocity is a vector, meaning it has both magnitude and direction. The "direction" of velocity is either positive or negative. Positive and negative velocities describe motion in opposite directions. In *Dance Dance Revolution*, negative would either be juking left or back and positive would be moving towards the forward and right arrows. However, if the player would rotate 90 degrees, the relative negative and positive directions might change. DDR would enter bazaar-o world, but it could happen. The context of a problem will tell you how to interpret positive and negative velocities for that problem.

### Sample Problems

1) Let *v*(*t*) be the velocity of a bug crawling back and forth on the *x*-axis. When *v*(*t*) is positive, the bug is moving to the right:

When *v*(*t*) is negative, the bug is moving to the left.

2) Calvin is biking away from his house with velocity *v*(*t*).

When *v*(*t*) is negative it means Calvin is biking back towards his house.

3) A bird is flying North with velocity *v*(*t*).

When *v*(*t*) is negative it means the bird is flying South.

**Speed** is the magnitude, or absolute value, of velocity. This means speed can't be negative!

### Sample Problem

Let *v*(*t*), in units per second, be the velocity of a bug crawling back and forth on the *x*-axis. When *v*(*t*) = -5 it means the bug is crawling left at a speed of 5 units per second.

When a question asks "how fast" something is going, you're being asked for the speed.

### Sample Problem

When Calvin's velocity is -7 miles per hour, how fast is he going?

Answer.

The question is asking for Calvin's speed, which is 7 miles per hour. The negative sign tells which direction Calvin is traveling, but doesn't have anything to do with his speed.

Now that we know what negative velocities mean, it's time to bring the integrals into it again and figure out what integrals of negative velocities mean.

When velocity is non-negative, we know that

When velocity is negative, the integral of velocity is also negative. We can think of this negative value as a **weighted distance**.

The number part tells the distance, and the negative sign tells the direction, which is the opposite of the direction travelled when *v*(*t*) is positive.

Take the area between *v*(*t*) and the *t*-axis that's above the axis. This is the total distance the bug travels to the right.

Take the area that's between *v*(*t*) and the *t*-axis, below the *t*-axis. This is the total distance the bug travels to the left.

Subtract:

(distance bug travels right) – (distance bug travels left) = 9 – 19 = -10.

This is the net change in the bug's position from time *t* = 0 to time *t* = 10.

We found this net change by taking a weighted sum of the areas between *v*(*t*) and the *t*-axis, which means we also just found the integral of *v*(*t*) from 0 to 10:

Whatever the units, when velocity is 0, speed is also 0. If something isn't moving at all, it's not moving in any direction. If *v*(*t*) is Calvin's velocity away from his house, measured in mph, then when *v*(*t*) = 0 Calvin isn't moving away from his house and he's not moving towards his house either.

#### 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 How high is the cat when |

#### Example 3

A squirrel starts two feet above ground and climbs a tree with velocity given by the graph below. When How high is the squirrel when |

#### 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 (a) Where on the number line is the bug when (b) At what time(s) does the bug change direction? (c) How many times during the interval 0 ≤ |

#### 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? |

#### Exercise 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?

#### Exercise 2

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

#### Exercise 3

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

#### Exercise 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?

#### Exercise 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?

#### Exercise 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?

#### Exercise 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?

#### Exercise 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.

#### Exercise 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?

#### Exercise 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?