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Quiz 5 MTH251.001 Spring 2021

Printed Name: ___________________________

You do not need to print out this quiz; you may provide your work and answers on your own separate paper.

Upload your solutions as a single PDF file to the โQuiz 5โ submission folder in our D2L course.

Your submission should be titled โ(your name) MTH251 Quiz 5.โ

Due Date: Your quiz results must be submitted by Friday May 21, by 11:59 p.m.

Instructions: – Show all your work; supporting work always needs to be included.

– Clearly indicate (circle or underline) your answer.

– For multiple choice questions, choice the single best answer.

__________________________________________________________________________________________

1.

Match the functions in graphs (A)-(D) with their derivatives (I)-(III) in figure 13. Explain why two of the

(2pt)

functions have the same derivative.

__________________________________________________________________________________________

2.

True or False.

a.

(2pt)

b.

c.

d.

๐

๐๐ฅ

๐

๐๐ฅ

๐

๐๐ฅ

๐

๐๐ฅ

๐ฅ โ21 = โ21๐ฅ โ21โ1

๐

๐

((๐ฅ 25 ) + (๐ฅ โ15 )) = ๐๐ฅ (๐ฅ 25 ) + ๐๐ฅ (๐ฅ โ15 )

๐ โ21 = โ21๐ โ21โ1

๐

๐

((๐ฅ 25 )(๐ฅ โ15 )) = ๐๐ฅ (๐ฅ 25 ) ๐๐ฅ (๐ฅ โ15 )

__________________________________________________________________________________________

3.

For each of the following find the equation of the indicated derivative.

(6pt)

2

a.

Given ๐ฆ = ๐ 10โ๐ฅ , find ๐ฆ โฒ .

b.

Given ๐(๐) = sin(2๐), find ๐ (4) (๐).

__________________________________________________________________________________________

4.

(5pt)

A suburban community receives its electrical power supply from a local power plant. An

engineer, after studying data on energy usage, determines that a good model for daily energy

usage between 6:00 a.m. and 8:00 p.m. is given by the function

๐กโ13 4

)

7

๐ธ(t) = 4 โ 2 (

for 6 โค ๐ก โค 20

where t = time in hours after midnight and E(t) is in megawatts (so t = 6 is 6:00 am, t = 12 is 12:00 pm

(noon) and so forth).

a. Determine ๐ธโฒ(๐ก).

b. Evaluate ๐ธโฒ(10).

Write a sentence explaining the meaning of the value you found for ๐ธโฒ(10). What is happening to

energy usage at 10 am?

__________________________________________________________________________________________

5.

Consider the relation:

(5pt)

๐ฆ2 โ ๐ฆ = ๐ฅ

The graph of which looks like:

a.

Use Implicit differentiation to find ๐ฆโฒ.

Since this graph is not a function, your final answer will likely contain both xโs and yโs.

b.

The point (2,-1) is on this graph (since (-1)2 – (-1) = 2).

Use your result in a) to find the equation of the tangent line at this point.

Quiz 5 MTH251.001 Spring 2021

Printed Name: ___________________________

You do not need to print out this quiz; you may provide your work and answers on your own separate paper.

Upload your solutions as a single PDF file to the โQuiz 5โ submission folder in our D2L course.

Your submission should be titled โ(your name) MTH251 Quiz 5.โ

Due Date: Your quiz results must be submitted by Friday May 21, by 11:59 p.m.

Instructions: – Show all your work; supporting work always needs to be included.

– Clearly indicate (circle or underline) your answer.

– For multiple choice questions, choice the single best answer.

__________________________________________________________________________________________

1.

Match the functions in graphs (A)-(D) with their derivatives (I)-(III) in figure 13. Explain why two of the

(2pt)

functions have the same derivative.

A:We can see on the graph A that the function represented is first increasing and then decreasing. This means

that her derivative has to be first positive and then negative.

This corresponds to the derivative III.

B:We can see on the graph B that the function represented is an increasing function. This so implies that her

derivative is only positive.

This so leads to the derivative I.

C:We can see on the graph C that the function represented is increasing then decreasing and again increasing.

This means that her derivative has to be positive then negative and positive again.

This so leads us to the derivative II.

D:We can see on the graph D that the function represented is first increasing and then decreasing. This is so the

same observation than for the graph A. This so means that the derivative of the graph D has to be positive and

then negative, just as the graph A.

We so have the derivative III which corresponds both for graph A and D

for the previous reasons.

__________________________________________________________________________________________

2.

(2pt)

True or False.

a.

๐

๐๐ฅ

๐ฅ

21

= 21๐ฅ

21โ1

TRUE because as the derivative of a function of the form is and that here we have

u = then is really

b.

๐

๐๐ฅ

((๐ฅ25) + (๐ฅโ15)) = ๐๐ฅ๐ (๐ฅ25) + ๐๐ฅ๐ (๐ฅโ15)

TRUE as the derivative of a sum is equal to the sum of the derivatives.

c.

๐

๐๐ฅ

๐

21

= 21๐

21โ1

FALSE because as the derivative of a function of the form is and that here we have

u = then should be 0 as uโ = 0 because .

d.

๐

๐๐ฅ

((๐ฅ25)(๐ฅโ15)) = ๐๐ฅ๐ (๐ฅ25) ๐๐ฅ๐ (๐ฅโ15)

FALSE as the derivative of a multiplication is not the multiplication of the

derivatives.

__________________________________________________________________________________________

3.

For each of the following find the equation of the indicated derivative.

(6pt)

2

10โ๐ฅ

‘

a.

Given ๐ฆ = ๐

, find ๐ฆ .

We have y(x) = and we are going to calculate yโ. As we have a function of the form then it means that its

derivative is going to be as . Here we have u = so uโ = .

So we can conclude that.

(4)

b.

Given ๐(ฮธ) = sin ๐ ๐๐ (2ฮธ) , find ๐ (ฮธ).

๐(๐)=sin(2๐) and we are going to calculate ๐(4)(๐) :

๐โ(๐)= 2cos(2๐) as the derivative of sin is cos

๐(2)(๐)= 2(-2sin(2๐)) = -4sin(2๐)

๐(3)(๐) = -4(2cos(2๐)) = -8cos(2๐)

๐(4)(๐) = -8(-2sin(2๐)) = 16sin(2๐)

so ๐(4)(๐) = 16sin(2๐)

__________________________________________________________________________________________

4.

A suburban community receives its electrical power supply from a local power plant. An

(5pt)

engineer, after studying data on energy usage, determines that a good model for daily energy

usage between 6:00 a.m. and 8:00 p.m. is given by the function

(

๐ธ(๐ก) = 4 โ 2

๐กโ13 4

7

)

for 6โค๐กโค20

where t = time in hours after midnight and E(t) is in megawatts (so t = 6 is 6:00 am, t = 12 is 12:00 pm

(noon) and so forth).

a. Determine ๐ธ'(๐ก).

We have and we are going to calculate E(t)โ :

So the derivative is

b. Evaluate ๐ธ'(10).

Write a sentence explaining the meaning of the value you found for ๐ธ'(10). What is happening to

energy usage at 10 am?

First, we calculate Eโ(10) :

MW/h

The derivative at a precise point is actually the gradient of the tangent from that point.

This means that here, we have calculated the gradient of the tangent at 10am. As this gradient is positive, this

means that at 10am the amount of energy is increasing by 0.09 MW/h.

__________________________________________________________________________________________

5.

Consider the relation:

(5pt)

2

๐ฆ โ๐ฆ=๐ฅ

The graph of which looks like:

a.

Use Implicit differentiation to find ๐ฆ’.

Since this graph is not a function, your final answer will likely contain both xโs and yโs.

We are going to find yโ while using the implicit differentiation

The derivative of y is so :

b.

The point (2,-1) is on this graph (since (-1)2 – (-1) = 2).

Use your result in a) to find the equation of the tangent line at this point.

In a) we have found that . Here we have find the equation of the tangent line at the point (2 ,-1). We are so

going to start by calculating the gradient of the tangent :

So from here we can deduce the equation of the tangent line, that we are going to call h, at the point (2, -1) :

So finally we have

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Exercice 3

We have y(x) = and we are going to calculate y’. As we have a function of the form then it means

that its derivative is going to be as . Here we have u = so u’ = .

So we can conclude that:

We have f(O)=sin(20) and we are going to calculate /)(O):

(O)=2cos(20) as the derivative of sin is cos

120)=2(-2sin(20)) = -4sin(20)

13)) = -4(2cos(20)) = -8cos(20)

(4)(O) = -86-2sin(20)) = 16 sin(20)

We finally so have /(“@) = 16 sin(20)

Exercice 4:

We have and we are going to calculate E(t)’:

So the derivative is

b) First, we calculate E'(10)

MW/h

The derivative at a precise point is actually the gradient of the tangent from that point.

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