# DERIVATIVES AS RATES OF CHANGE WORKSHEET

Problem 1 :

The temperature T in celsius in a long rod of length 10 m, insulated at both ends is the function of length x given by

T = x(10 - x)

Prove that the rate of change of temperature at the midpoint of the rod is 0.

Solution

Problem 2 :

A person learnt 100 words for an English test. The number of words the person remembers in t days after learning is given by

W(t) = 100×(1− 0.1t)2 , 0  t 10

What is the rate at which the person forgets the words 2 days after learning?

Solution

Problem 3 :

A particle moves so that the distance moved is according to the law

s(t) = t3/3 - t2 + 3

At what time the velocity and acceleration is zero.

Solution

Problem 4 :

A particle is fired straight up from the ground to reach a height of s feet in t seconds where

s(t) = 128t - 16t2

i) Compute the maximum height of the particle reached ?

ii)  What is the velocity when the particle hits the ground ?

Solution

Problem 5 :

A particle moves along a horizontal line such that its position at any time t ≥ 0 is given by

s(t) = t3 − 6t2 + 9t + 1

where s is measured in meters and t in seconds?

(i) At what time the particle is at rest?

(ii) At what time the particle changes its direction?

(iii) Find the total distance travelled by the particle in the first 2 seconds.

Solution

1)  So, temperature at the midpoint of the rod is 0.

2)   the number of words he forgets is 16.

3) t = 1

4) i)  256 ft    ii)  The velocity when it reaches the ground is -128 ft/s

5)  i)  t = 1 and t = 3

ii)  Then the particle changes its direction in between 1 to 3 seconds.

iii)  6 meters

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