This document as the title suggests contains exam questions and their respective solutions. I had the privilege to take this exam last year and I realized this may be an extremely useful resource for anyone who needs more preparation material for their exams.
Math 211, Multivariable Calculus, Fall 2011
Final Exam Solutions
1. (10 points) Find the equation of the plane that contains both the point (−1, 1, 2) and
the line given by
x = 1 − t, y = 1 + 2t, z = 2 − 3t.
Solutions: A point on the line is (1, 1, 2) and a vector parallel to the line is h−1, 2, −3i.
Another vector parallel to the plane we want is h2, 0, 0i (the vector between the two
points we have) so a vector perpendicular to the plane is given by
n = h2, 0, 0i × h−1, 2, −3i = h0, 6, 4i .
The equation of the plane is therefore
6(y − 1) + 4(z − 2) = 0
or
6y + 4z = 14 .
2. (10 points) Consider the function f (x, y) = x2 (y − 1).
(a) What is the directional derivative of f at (1, 3) in the direction of the vector
v = h3, −4i?
Solution: ∇f = h2x(y − 1), x2 i, so ∇f (1, 3) = h4, 1i. The directional derivatives
is therefore
h4, 1i · h3, −4i 8 8
=√ = .
| h3, −4i | 25 5
(b) What is the maximum directional derivative of f at (1, 3), and in which direction
does it occur?
Solution: The maximum directional derivative is the length of the gradient vector
√
17 and it is in the direction of the gradient vector h4, 1i .
3. (10 points) Find the linear approximation to the function f (x, y) = 2 − sin(−x − 3y)
at the point P = (0, π), and then use your answer to estimate f (0.001, π).
Solution: The linear approximation is given by
l(x, y) = f (0, π) + fx (0, π)(x − 0) + fy (0, π)(y − π).
We have fx = cos(−x − 3y), so fx (0, π) = cos(−3π) = −1 and fy = 3 cos(−x − 3y), so
fy (0, π) = 3 cos(−3π) = −3. Also f (0, π) = 2 − sin(−3π) = 2. Therefore
l(x, y) = 2 − (x − 0) − 3(y − π) .
So we have
f (0.001, π) ≈ 1.999 .
, 4. (5 points) Prove that, for any curve described by a vector-valued function r(t), the
unit tangent vector T(t) is always orthogonal to its derivative T0 (t).
Solution: The unit tangent vector is a unit vector so
T(t) · T(t) = 1.
Differentiating both sides we get
T0 (t) · T(t) + T(t) · T0 (t) = 0
so
T0 (t) · T(t) = 0.
Therefore T0 (t) is perpendicular to T(t).
5. (10 points) Let C be the curve given by
r(t) = (cos t + t sin t)i + (sin t − t cos t)j, for t > 0.
Find the unit tangent vector T(t), unit normal vector N(t), and curvature κ(t) for C.
(Your answers should be functions of t.)
Solution: We have
r0 (t) = ht cos t, t sin ti
so p √
|r0 (t)| = t2 cos2 t + t2 sin2 t = t2 = t
since t > 0. Therefore the unit tangent vector is
r0 (t)
T(t) = = hcos t, sin ti .
|r0 (t)|
Differentiating this we get
T0 (t) = h− sin t, cos ti .
Then
T0 (t) h− sin t, cos ti
N(t) = 0
= = h− sin t, cos ti .
|T (t)| 1
The curvature is then
|T0 (t)| 1
κ(t) = 0 = .
|r (t)| t
6. (5 points) Show that the function
2 2
x − y if (x, y) 6= (0, 0);
f (x, y) = x2 + y 2
0 if (x, y) = (0, 0);
is not continuous at (0, 0).
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller radhikakhatri. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.49. You're not tied to anything after your purchase.
