preTEST2A Solution MAT225 Multivariable Calculus

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RVC/RVE (Question 1) Product Rule
Find the max of f(x) = sin(x),
e−x ε[0, ].
x π
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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
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(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?
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(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.
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(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.
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(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.
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(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).
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(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.
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(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).
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(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
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(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.
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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)
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