This document appears to be a math test containing 8 multi-part questions testing various calculus concepts such as: product rule, quotient rule, optimization, Lagrange multipliers, chain rule, implicit differentiation, and a reference sheet of important derivatives. The test covers topics including finding maxima/minima, normal lines, vectors, work, investments, gradients, tangent planes, and conic sections.

1. MAT225 TEST2A Name:
Show all work algebraically if possible.
RVC/RVE (Question 1) Product Rule
Find the max of f(x) = sin(x),
e−x ε[0, ].
x π
TEST2A page: 1

2. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 2

3. MAT225 TEST2A Name:
Show all work algebraically if possible.
RVD/RVE (Question 2) Quotient Rule
Find the equation of the Normal Line to g(x) at x = 1.5 given g(x) = .
1
1+9e−x
TEST2A page: 3

4. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 4

5. MAT225 TEST2A Name:
Show all work algebraically if possible.
(1) Let ​u​ = <1,2,3>, ​v​ = <0,–2,5>, ​w​ = <1,1,0>
(1a) Find ​v​ x ​u
(1b) Find ​w​(​v​ x ​u​)
(1c) Find |​v​ x ​u​|
(1) Let ​u​ = <1m,2m,3m>, ​v​ = <0m,–2m,5m>, ​w​ = <1m,1m,0m>
(1d) What are the units of |​v​ x ​u​| and why?
(1) Let ​u​ = <1yd,2yd,3yd>, ​v​ = <0yd,–2yd,5yd>, ​w​ = <1yd,1yd,0yd>
(1e) What are the units of ​w​(​v​ x ​u​) and why?
TEST2A page: 5

6. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 6

7. MAT225 TEST2A Name:
Show all work algebraically if possible.
(2) Work
A tractor pulls a log 2500 feet and the tension in the cable connecting the
tractor to the log is 2600 pounds. The angle between the force and displacement
vectors is 35º.
Work is defined as W = ​F​ • ​s​ where ​F​ is the force vector and ​s​ the
displacement vector.
(2a) Calculate W in polar form.
(2b) Calculate W in cartesian form.
TEST2A page: 7

8. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 8

9. MAT225 TEST2A Name:
Show all work algebraically if possible.
(3) Investments:
Let x = an amount invested in AA rated bonds at 6.5%,
Let y = an amount invested in A rated bonds at 7%,
Let z = an amount invested in B rated bonds at 9%.
If you decide to invest twice as much in B bonds as in A, your investment strategy is
described by the following system of equations:
x + y + z = total investment
0.065x + 0.07y + 0.09z = desired return
0x + 2y z
− = 0
Let your total investment equal US$45,000.00 and your desired return equal
US$3,000.00.
(3a) Given AX = B, find det(A).
(3b) Given AX = B, find x using Cramer’s Rule.
TEST2A page: 9

10. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 10

11. MAT225 TEST2A Name:
Show all work algebraically if possible.
(3) Investments:
Let x = an amount invested in AA rated bonds at 6.5%,
Let y = an amount invested in A rated bonds at 7%,
Let z = an amount invested in B rated bonds at 9%.
If you decide to invest twice as much in B bonds as in A, your investment strategy is
described by the following system of equations:
x + y + z = total investment
0.065x + 0.07y + 0.09z = desired return
0x + 2y z
− = 0
Let your total investment equal US$45,000.00 and your desired return equal
US$3,000.00.
(3c) Given AX = B, find y using Cramer’s Rule.
(3d) Given AX = B, find z using Cramer’s Rule.
TEST2A page: 11

12. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 12

13. MAT225 TEST2A Name:
Show all work algebraically if possible.
(4) Gradients
Given the surface y z 3
x3 + 2 =
(4a) Let f(x,y,z)= . Find the Gradient of f(x,y,z) at P(-1,1,2).
y z 3
x3 + 2 −
(4b) Derive the Tangent Plane equation to the given surface at P(-1,1,2).
(4c) Use a linear approximation to approximate the value of f(-1.1,1.1,1.9).
TEST2A page: 13

14. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 14

15. MAT225 TEST2A Name:
Show all work algebraically if possible.
(5) Optimization
Let f(x,y) = x y x y 6
3 2 + 2 2 − 6 − 4 + 1
(5a) Find the critical points of f(x,y) in the first quadrant.
(5b) Use the 2nd Partials Test to classify the nature of your critical point.
(5c) Calculate the critical value of f(x,y) in the first quadrant.
TEST2A page: 15

16. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 16

17. MAT225 TEST2A Name:
Show all work algebraically if possible.
(6) LaGrange Multipliers
Maximize the volume V = xyz, subject to the constraint: .
x2 + y2 + z = 1
(6a) Write the LaGrange Multiplier equations to maximize V.
(6b) Solve your equations (still assuming x > 0 and y > 0).
TEST2A page: 17

18. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 18

19. MAT225 TEST2A Name:
Show all work algebraically if possible.
(7) Chain Rule
Let w = and x =​ , y = cos(t).
x
y2
− x2
et
(7a) Find w’(t) when t = using the new chain rule.
2
π
(7b) Find w’(t) when t = directly.
2
π
TEST2A page: 19

20. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 20

21. MAT225 TEST2A Name:
Show all work algebraically if possible.
(8) Chain Rule
Given z = f(x,y) = 0 and y = g(x), then​ .
δf
δx dx
dx
+
δf
δy dx
dy
= 0 ⇒ dx
dy
= δf
δy
δx
−δf
(8) Given the conic section 9 x y 8x 6y 1 ,
2
+ 4 2
− 1 + 1 − 1 = 0
(8a) Find using the result given above.
dx
dy
(8b) Check your answer using Elementary Calculus.
TEST2A page: 21

22. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 22

23. MAT225 TEST2A Name:
Show all work algebraically if possible.
Reference Sheet: Derivatives You Should Know Cold!
Power Functions:
x nx
d
dx
n = n−1
Trig Functions:
sin(x) os(x)
d
dx
= c cos(x) in(x)
d
dx
= − s
tan(x) (x)
d
dx
= sec2 cot(x) (x)
d
dx
= − csc2
sec(x) ec(x) tan(x)
d
dx
= s csc(x) sc(x) cot(x)
d
dx
= − c
Transcendental Functions:
e
d
dx
x = ex a n(a) a
d
dx
x = l x
ln(x)
d
dx = x
1
log (x)
d
dx a = 1
ln(a) x
1
Inverse Trig Functions:
sin (x)
d
dx
−1
= 1
√1−x2
cos (x)
d
dx
−1
= −1
√1−x2
tan (x)
d
dx
−1
= 1
1+x2 cot (x)
d
dx
−1
= −1
1+x2
Product Rule:
f(x) g(x) (x) g (x) (x) f (x)
d
dx = f ′ + g ′
Quotient Rule:
d
dx
f(x)
g(x) = g (x)
2
g(x) f (x) − f(x) g (x)
′ ′
Chain Rule:
f(g(x)) (g(x)) g (x)
d
dx = f′ ′
Difference Quotient:
f’(x) =​ lim
h→0
h
f(x+h) − f(x)
TEST2A page: 23

24. MAT225 TEST2A Name:
Show all work algebraically if possible.
TEST2A page: 24