MATH 104-184 Quiz #3
October 22
Grade:
First Name:
Last Name:
Student-No:
Section:
Very short answer questions
1. 2 marks Each part is worth 1 mark. Please write your answers in the boxes.
scheme: 1 for each correct, 0 otherwise
Marking
(a) Differentiate y = 2x cos x.
Answer: y = 2x (ln 2) cos x − 2x sin x
Solution: Just use the Product Rule.
(b) The cost function is given by C(q) = q 2 + 2q + 3. Estimate the cost of producing the 10th
unit.
Answer: 20
Solution: M C(q) = C 0 (q) = (2q + 2). Thus, M C(9) = 20.
Short answer questions — you must show your work
2. 4 marks Each part is worth 2 marks.
(a) Find the derivative of the function g(x) = cos[sin(x3 )].
Answer: −3x2 cos(x3 ) sin[sin(x3 )]
Solution: Marking scheme: 1pt for a decent step towards the first line of differentiation. 2pt for a correct final answer (or near perfect solution)
g(x) = cos[sin(x3 )]
d sin(x3 )
dx
d 3
g 0 (x) = − sin[sin(x3 )] · cos(x3 ) ·
x
dx
g 0 (x) = − sin[sin(x3 )] · cos(x3 ) · 3x2
g 0 (x) = − sin[sin(x3 )] ·
(b) Find the slope of the tangent line to 2x3 + 4xy + 2y 3 = 8 at the point (1, 1)
Answer: −1
Solution: Marking scheme: 1pt for a decent step towards the first line of differentiation. 2pt for a correct final answer (or near perfect solution)
We differentiate implicitly to get
6x2 + 4y + 4xy 0 + 6y 2 y 0 = 0
Subbing in x = 1 and y = 1 yields
6 + 4 + 4y 0 + 6y 0 = 0
from which we see that 10y 0 = −10 and y 0 = −1.
Long answer question — you must show your work
2
3. 4 marks Determine where the tangent line to h(x) = xln(x ) is horizontal.
Answer: x = 1
Solution: Marking scheme: 1pt for taking the log of both sides; 1pt differentiating
properly; 1pt for setting h’=0; 1pt for getting x = 1.
Use logarithmic differentiation:
2
h(x) = xln(x )
h
i
ln(x2 )
ln h = ln x
ln h = ln(x2 ) · ln x
ln h = 2 ln x · ln x
ln h = 2(ln x)2
1
1 0
· h = 4 ln x ·
h
x
4
ln
x
h0 = h ·
x
From here, we note that h 6= 0 since x = 0 is not in the domain (we can’t put x = 0 into
ln x2 ). This means that h0 (x) = 0 if and only if ln x = 0. Therefore, x = 1 is the only
solution.
MATH 104-184 Quiz #3
October 22
Grade:
First Name:
Last Name:
Student-No:
Section:
Very short answer questions
1. 2 marks Each part is worth 1 mark. Please write your answers in the boxes.
scheme: 1 for each correct, 0 otherwise
Marking
(a) Differentiate y = 3x sin x.
Answer: y = 3x (ln 3) sin x + 3x cos x
Solution: Just use the Product Rule.
(b) The cost function is given by C(q) = q 2 + 3q + 2. Estimate the cost of producing the 12th
unit.
Answer: 25
Solution: M C(q) = C 0 (q) = (2q + 3). Thus, M C(11) = 25.
Short answer questions — you must show your work
2. 4 marks Each part is worth 2 marks.
(a) Find the derivative of the function g(x) = cos[cos(x3 )].
Answer: 3x2 sin(x3 ) sin[cos(x3 )]
Solution: Marking scheme: 1pt for a decent step towards the first line of differentiation. 2pt for a correct final answer (or near perfect solution)
g(x) = cos[cos(x3 )]
g 0 (x) = − sin[cos(x3 )] ·
d cos(x3 )
dx
g 0 (x) = − sin[cos(x3 )] · (− sin(x3 )) ·
d 3
x
dx
g 0 (x) = sin[cos(x3 )] · sin(x3 ) · 3x2
(b) Find the slope of the tangent line to 2x3 + 3xy + 2y 3 = 7 at the point (1, 1)
Answer: −1
Solution: Marking scheme: 1pt for a decent step towards the first line of differentiation. 2pt for a correct final answer (or near perfect solution)
We differentiate implicitly to get
6x2 + 3y + 3xy 0 + 6y 2 y 0 = 0
Subbing in x = 1 and y = 1 yields
6 + 3 + 3y 0 + 6y 0 = 0
from which we see that 9y 0 = −9 and y 0 = −1.
Long answer question — you must show your work
3
3. 4 marks Determine where the tangent line to h(x) = xln(x ) is horizontal.
Answer: x = 1
Solution: Marking scheme: 1pt for taking the log of both sides; 1pt differentiating
properly; 1pt for setting h’=0; 1pt for getting x = 1.
Use logarithmic differentiation:
3
h(x) = xln(x )
h
i
ln(x3 )
ln h = ln x
ln h = ln(x3 ) · ln x
ln h = 3 ln x · ln x
ln h = 3(ln x)2
1
1 0
· h = 6 ln x ·
h
x
6
ln
x
h0 = h ·
x
From here, we note that h 6= 0 since x = 0 is not in the domain (we can’t put x = 0 into
ln x3 ). This means that h0 (x) = 0 if and only if ln x = 0. Therefore, x = 1 is the only
solution.
MATH 104-184 Quiz #3
October 22
Grade:
First Name:
Last Name:
Student-No:
Section:
Very short answer questions
1. 2 marks Each part is worth 1 mark. Please write your answers in the boxes.
scheme: 1 for each correct, 0 otherwise
Marking
(a) Differentiate y = 4x cos x.
Answer: y = 4x (ln 4) cos x − 4x sin x
Solution: Just use the Product Rule.
(b) The cost function is given by C(q) = q 2 + 2q + 3. Estimate the cost of producing the 13th
unit.
Answer: 26
Solution: M C(q) = C 0 (q) = (2q + 2). Thus, M C(12) = 26.
Short answer questions — you must show your work
2. 4 marks Each part is worth 2 marks.
(a) Find the derivative of the function g(x) = sin[cos(x3 )].
Answer: −3x2 sin(x3 ) cos[cos(x3 )]
Solution: Marking scheme: 1pt for a decent step towards the first line of differentiation. 2pt for a correct final answer (or near perfect solution)
g(x) = sin[cos(x3 )]
g 0 (x) = cos[cos(x3 )] ·
d cos(x3 )
dx
d 3
x
dx
g 0 (x) = − cos[cos(x3 )] · sin(x3 ) · 3x2
g 0 (x) = cos[cos(x3 )] · (− sin(x3 )) ·
(b) Find the slope of the tangent line to 5x3 + 3xy + 2y 3 = 10 at the point (1, 1)
Answer: −2
Solution: Marking scheme: 1pt for a decent step towards the first line of differentiation. 2pt for a correct final answer (or near perfect solution)
We differentiate implicitly to get
15x2 + 3y + 3xy 0 + 6y 2 y 0 = 0
Subbing in x = 1 and y = 1 yields
15 + 3 + 3y 0 + 6y 0 = 0
from which we see that 9y 0 = −18 and y 0 = −2.
Long answer question — you must show your work
4
3. 4 marks Determine where the tangent line to h(x) = xln(x ) is horizontal.
Answer: x = 1
Solution: Marking scheme: 1pt for taking the log of both sides; 1pt differentiating
properly; 1pt for setting h’=0; 1pt for getting x = 1.
Use logarithmic differentiation:
4
h(x) = xln(x )
h
i
ln(x4 )
ln h = ln x
ln h = ln(x4 ) · ln x
ln h = 4 ln x · ln x
ln h = 4(ln x)2
1
1 0
· h = 8 ln x ·
h
x
8
ln
x
h0 = h ·
x
From here, we note that h 6= 0 since x = 0 is not in the domain (we can’t put x = 0 into
ln x3 ). This means that h0 (x) = 0 if and only if ln x = 0. Therefore, x = 1 is the only
solution.
MATH 104-184 Quiz #3
October 22
Grade:
First Name:
Last Name:
Student-No:
Section:
Very short answer questions
1. 2 marks Each part is worth 1 mark. Please write your answers in the boxes.
scheme: 1 for each correct, 0 otherwise
Marking
(a) Differentiate y = 5x cos x.
Answer: y = 5x (ln 5) cos x − 5x sin x
Solution: Just use the Product Rule.
(b) The cost function is given by C(q) = q 2 + 2q + 34. Estimate the cost of producing the
17th unit.
Answer: 34
Solution: M C(q) = C 0 (q) = 2q + 2. Thus, M C(16) = 34.
Short answer questions — you must show your work
2. 4 marks Each part is worth 2 marks.
(a) Find the derivative of the function g(x) = sin[sin(x3 )].
Answer: 3x2 cos(x3 ) cos[sin(x3 )]
Solution: Marking scheme: 1pt for a decent step towards the first line of differentiation. 2pt for a correct final answer (or near perfect solution)
g(x) = sin[sin(x3 )]
d sin(x3 )
dx
d 3
g 0 (x) = cos[sin(x3 )] · (cos(x3 )) ·
x
dx
g 0 (x) = cos[sin(x3 )] · cos(x3 ) · 3x2
g 0 (x) = cos[sin(x3 )] ·
(b) Find the slope of the tangent line to 2x3 + 7xy + 2y 3 = 12 at the point (1, 1)
Answer: −1
Solution: Marking scheme: 1pt for a decent step towards the first line of differentiation. 2pt for a correct final answer (or near perfect solution)
We differentiate implicitly to get
6x2 + 7y + 7xy 0 + 6y 2 y 0 = 0
Subbing in x = 1 and y = 1 yields
6 + 7 + 7y 0 + 6y 0 = 0
from which we see that 13y 0 = −13 and y 0 = −1.
Long answer question — you must show your work
5
3. 4 marks Determine where the tangent line to h(x) = xln(x ) is horizontal.
Answer: x = 1
Solution: Marking scheme: 1pt for taking the log of both sides; 1pt differentiating
properly; 1pt for setting h’=0; 1pt for getting x = 1.
Use logarithmic differentiation:
5
h(x) = xln(x )
h
i
ln(x5 )
ln h = ln x
ln h = ln(x5 ) · ln x
ln h = 5 ln x · ln x
ln h = 5(ln x)2
1
1 0
· h = 10 ln x ·
h
x
10
ln
x
h0 = h ·
x
From here, we note that h 6= 0 since x = 0 is not in the domain (we can’t put x = 0 into
ln x5 ). This means that h0 (x) = 0 if and only if ln x = 0. Therefore, x = 1 is the only
solution.