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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

Example 1

Suppose Jen's velocity in mph was measured every ten minutes for one hour, and that her velocity was decreasing over that hour. The recorded values are shown in the table below. Use a right-hand sum to estimate how far Jen travelled during that hour. Is this an over- or under-estimate of the distance she really travelled?

Example 2

The table below gives the velocity of a tortoise over nine minutes with measurements taken every 3 minutes.

Estimate the distance the tortoise travels during these nine minutes, using

a) a left-hand sum

b) a right-hand sum

c) a trapezoid sum

Example 3

The table below gives the velocity of a dinosaur over thirty seconds, with measurements taken every 5 seconds.

Estimate the distance the dinosaur travels during these thirty seconds, using

a) a left-hand sum

b) a right-hand sum

c) a trapezoid sum

Example 4

The velocity of a garden snail over 3 minutes, measured twice per minute, is given in the table below.

Estimate the distance the snail travels during these three minutes.

Example 5

The velocity of a cheetah over one minute, measured every 10 seconds, is given in the table below.

Estimate the distance the cheetah travels during this minute.

Example 6

Use the trapezoid rule and the table below to estimate how far the cheetah travels in one minute.

Example 7

The velocity of a pink elephant, in feet per minute, is given by the function v(t) below.

How far does the elephant travel

(a) from t = 0 to t = 10?

(b) from t = 10 to t = 25?

(c) from t = 25 to t = 30?

(d) from t = 0 to t = 30?

Example 8

A bug starts out at position -4 on a number line and crawls along with velocity v(t) units per second, where v(t) is given by the graph below.

Where is the bug when

(a) t = 2?

(b) t = 3?

(c) t = 5?

Example 9

A cat is climbing a tree. The cat is one foot above ground when its owner starts a stopwatch. The cat's velocity, in feet per second, is given by the following graph:

How high in the tree is the cat when the stopwatch reads

(a) 3 seconds?

(b) 6 seconds?

(c) 8 seconds?

(d) 10 seconds?

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