MATH 1320 : Spring 2014
Lab 11
Lab Instructor : Kurt VanNess
Name:
Score:
Write all your solutions on a separate sheet of paper.
1. Let f (x, y) =
p
x2 + 3y 2 .
(a) Find fx and fy at (1, 4).
(b) Write an equation of the tangent plane to the graph of f at (1, 4).
(c) Make an approximation for (1.1, 3.9).
(d) What is the actual value?
(e) What is the error in your approximation?
2. If R is the total resistance of three resistors, connected in parallel, with resistances R1 , R2 , R3 , then
1
1
1
1
=
+
+
.
R
R1
R2
R3
If the resistances are measured in ohms as R1 = 25Ω, R2 = 35Ω, and R3 = 50Ω, with a possible error
of 0.5% in each case, estimate the maximum error in the calculated value of R.
3. Wheat production W in a given year depends on the average temperature T and the annual rainfall
R. Scientists estimate that the average temperature is rising at a rate of 0.15◦ C/year and rainfall is
decreasing at a rate of 0.1 cm/year. They also estimate that, at current production levels, ∂W/∂T = −2
and ∂W/∂R = 6.
(a) What is the significance of the signs of these partial derivatives?
(b) Estimate the current rate of change of wheat production, dW/dt.
4. If z = f (x, y) where x = r cos θ, y = r sin θ (the polar coordinates):
(a) Find ∂z/∂r and ∂z/∂θ.
(b) Show that
∂z
∂x
2
+
∂z
∂y
2
=
∂z
∂r
2
+
1
r2
∂z
∂θ
2
.
5. If f and g are twice differentiable functions of a single variable, show that
u(x, t) = f (x + at) + g(x − at)
is a solution to the wave equation
2
∂2u
2∂ u
=
a
.
∂t2
∂x2
[Hint: Introduce variables v = x + at and w = x − at.]
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