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You CAN Ace Calculus

17calculus > exam list > calc3 exam A2

 partial derivatives gradients directional derivatives lagrange multipliers tangent plane multi-variable min/max second partial derivative test

### Calculus Main Topics

Single Variable Calculus

Multi-Variable Calculus

### Tools

math tools

general learning tools

other exams from this semester

exam A1 - exam A2 - exam A3 - exam A4

complete exam list

Calculus 3 - Exam 2 ( Semester A )

Exam Overview

This is the second exam for third semester (multi-variable) calculus. This exam contains 7 questions, one basic partial derivative, one chain rule, two directional derivatives, one lagrange multiplier, one multi-variable min/max and one tangent plane.

Exam Details

Tools

Time

2 hours

Calculators

no

Questions

7

Formula Sheet(s)

1 page, 8.5x11 or A4

Total Points

75

Other Tools

none

Instructions:
- For each problem, correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
- Correct notation counts (i.e. points will be taken off for incorrect notation).
[ Click on the question to reveal/hide the solution. ]

Question 1

(5 points) Calculate the first partial derivatives of $$f(u,v,t)=e^{uv}\sin(ut)$$.

solution

Question 2

(10 points) If $$u=x^2-2y^2+z^3$$ and $$x=\sin(t), y=e^t, z=3t$$, calculate $$du/dt$$ using the appropriate chain rule. Make sure your final answer has only numbers and t (no x, y or z).

solution

Question 3

(10 points) Calculate the directional derivative of $$f(x,y,z)=x^3+y^3z$$ at $$(-1,2,1)$$ in the direction toward the point $$(0,3,3)$$. Is this the maximum directional derivative? Explain and show work why or why not.

solution

Question 4

(15 points) The directional derivative of $$z=f(x,y)$$ at $$(2,1)$$ in the direction toward the point $$(1,3)$$ is $$-2/\sqrt{5}$$ and the directional derivative in the direction toward the point $$(5,5)$$ is 1. Compute $$\partial z / \partial x$$ and $$\partial z / \partial y$$ at $$(2,1)$$.

solution

Question 5

(10 points) Use Lagrange multipliers to find the point(s) on the surface $$z^2-x^2-y^2=0$$ that are closest to the point $$(1,7,0)$$.

solution

Question 6

(10 points) Find the point(s) on the surface $$z^2-x^2-y^2=0$$ that are closest to the point $$(1,7,0)$$ by calculating the critical points and using the Second Derivative Partials Test to conclude that your answer(s) are minimums.

(10 points) Find the equation of the tangent plane and the parametric equations of the normal line to the surface $$f(x,y)=x^2y$$ at the point $$(2,1,4)$$.