# Rate of Change of Function - Multivariable Calculus - Past Paper, Exams for Calculus. Agra University

## Calculus

Description: These are the notes of Past Paper of Multivariable Calculus. Key important points are: Rate of Change of Function, Trigonometric Identities, Spherical Coordinates, Approximate Value, Numerical Values of Derivatives, Critical Points of Function, Tangent Plane
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Name:
Lab Section:
MATH 215 – Fall 2004
FINAL EXAM
Show your work in this booklet.
Do NOT submit loose sheets of paper–They won’t be graded
Problem Points Score
1 15
2 10
3 25
4 10
5 15
6 15
7 10
TOTAL 100
Some useful trigonometric identities:
sin2θ+ cos2θ= 1 cos 2θ= cos2θsin2θsin 2θ= 2 sin θcos θ
sin2θ=1cos 2θ
2cos2θ=1 + cos 2θ
2
Spherical coordinates:
x=ρcos(θ) sin(φ)y=ρsin(θ) sin(φ)z=ρcos(φ)
1
Problem 1. (15 points) This problem is about the function
f(x, y, z) = 3zy + 4xcos(z).
(a) What is the rate of change of the function of fat (1,1,0) in the direction from this point to
the origin?
(b) Give an approximate value of f(0.9,1.2,0.11).
CONTINUED ON THE NEXT PAGE
2
(c) Recall that f(x, y, z) = 3zy + 4xcos(z).
The equation f(x, y, z) = 4 implicitly deﬁnes zas a function of (x, y), if we agree that z= 0 if
(x, y) = (1,1).
Find the numerical values of the derivatives
z
x(1,1) and z
y (1,1).
3
Problem 2. (10 points)
(a) Find and classify the critical points of the function f(x, y) = 2x2+ 8xy 9y2+ 4y4.
(b) Find the equation of the tangent plane to the surface z+ 2x28xy + 9y24y= 0 at the
point (2,0,8).
4