NAME:
Test #1b
MA3160, T. Olson, Spring 2019
Please show work or give reasoning for every answer. (No credit will be given for correct answers without an indication of
how you arrived at your conclusion.)
If you obtain an answer or part of an answer with your calculator, please indicate what you punched into your calculator
and what the output was.
If you use a formula, please write down the formula that you are using.
1. The graph of a function of two variables, z = f (x, y), is shown.
The dot shown on the graph is (0, 3, 2).
Answer the following questions based on this graph.
(a) Sketch a graph of the trace of f with y = 3.
Label your axes with “x” and/or “y” and/or “z.”
(b) What is the sign of the partial derivative fy (0, 3)?
circle one:
How can you tell?
positive
negative
2. Ida Nonuttin was asked to find the equation of the plane tangent to the surface z = 3x2
the point where (x, y) = (2, 1). Ida’s answer was
z = (6x + 2y)(x
2) + ( 12y 2 + 2x)(y
1) + 12.
(a) At a glance, how do you know this is not the equation for the tangent plane?
(b) Write the correct equation for the tangent plane.
1
about zero
4y 3 + 2xy at
@z
.
3. (a) Suppose z = f (x, y), where x = g(s, t, u) and y = h(s, t, u). Write the general chain rule for
@u
Write your answer in terms of partial derivatives of f , g, and h.
@z
at the point (s, t, u) = (4, 4, 3),
@u
x = u sin(s t) and y = st + us .
(b) Use your answer from part (a) to find
when
z = x3 + x2 y
with
4. Use the level curves for h(x, y) to answer the following.
The square is placed at (1.5, 0).
The little dot is placed at (1, 1)
(a) At the point (1.5, 0), is the partial derivative
positive or negative or close to zero?
@h
@y
What feature of the contour plot tells you this?
(b) Now look at the little dot, at the point (1, 1). In what direction does the gradient point at (1, 1)?
Draw an arrow in the center of the contour graph showing the direction of gradh(1, 1) and write a
sentence below describing the relationship between the level curve and the gradient.
2
5. Given the following information about a function f (x, y) and its derivatives near the point (x, y) = (2, 3),
f (2, 3) = 7,
fx (2, 3) = 8,
fy (2, 3) = 9,
(a) estimate the value of f (1.8, 3.1), showing how you arrive at your answer.
(b) Find rf (2, 3), the gradient of f .
(c) Compute the directional derivative of f at the point (2, 3) in the direction 2~ı + ~|.
(d) Identify a vector which is perpendicular to the graph of z = f (x, y) at the point (2, 3, 7).
(e) Suppose f is measured in miles, x is measured in hours, and y is measured in gallons. In the following,
fill in the blanks with the UNITS for each number.
f (2, 3) = 7
,
fx (2, 3) = 8
3
,
fy (2, 3) = 9
,