The equation for the length of a hypotenuse of a right

| August 30, 2017

Question
Problems 1-6 are each worth 15 points. Problem 7 is worth 10 points. Please show all work.
Failure to show intermediate steps will result in zero credit.
1. The equation for the length of a hypotenuse of a right triangle is given by the function z
below. Please show all intermediate steps.
a) Find the total differential equation for z
b) Determine the value of the total differential (dz) when x = 6, y = 8, dx = 0.25 and
dy = -0.125 (all values are in inches)
c) Interpret your answer from question b.
z x 2 y2

2. Use calculus to sketch the following equation. Use the second derivative test to determine
whether a stationary point is a maximum, minimum or inflection point. Please show all
intermediate steps.

y

x2 – 3
x3

3. Find dz/dt given the following information. Please show all intermediate steps.
z

ln(x 2 y 2 )
t

where x e – t
y e t

4. Find all local stationary points. Determine whether each point is a maximum, minimum or a
saddle point. Please show all intermediate steps.

z 3×2 y – 2xy 2 5y2

5. Use the Lagrange multiplier method to find stationary points for the following problem.
Evaluate the second order conditions. Please show all intermediate steps.
z 28x – 2x 2 6xy – 5y2 10y
subject to 2x 10y 250

6. Evaluate the following indefinite integral. Please show all intermediate steps.
y 5lnx * lnx dx

7. Evaluate the following definite integral. Please show all intermediate steps.
1

(4x

0

3

2x 2 8x 6) π (3x 2 x 2) dx

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