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Math 2224 Test 1: Calculus Problems, Exams of Calculus

Virginia Polytechnic Institute and State University (Virginia Tech)Calculus

A calculus test from math 2224, including problems on directional derivatives, tangent lines, level curves, and linearization. Students are required to find derivatives, equations of tangent lines, and level curves, as well as understand the relationship between gradients and level curves. The document also includes problems on trigonometric functions and vector calculus.

Typology: Exams

Pre 2010

Uploaded on 09/23/2008

dowdy
dowdy ๐Ÿ‡บ๐Ÿ‡ธ

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Math 2224 Test 1 Name__________Solution__________________
September 20, 2007 (pledged)
All work must be shown. Credit will not be given, even for correct final answers, unless all
the correct and necessary calculations, procedures, and/or reasoning leading to such
answers are shown.
Calculators should not be needed and may not be used on this test.
For numbers 1 - 6, please do all work on your own paper. Nothing on this page will be
graded. You do not have to write the answers on this page. For number 7, matching, you
should write the answers in the appropriate blank. When you are finished staple this page
to the front of your work pages.
1. Find the directional derivative for the function 2
3
(, )
x
y
fxy yx
=
+ at the point (1,2)Pโˆ’ in the
direction of the point (2, 2)Qโˆ’. What is the maximum value of the directional derivative at P?
21 3
2
423
(, )
23 1
(, ) , (, )
26 1 1 5
(1,2) 7, (1,2)
21 4 1 4
5
(1,2) 7,
4
34
2 (1),2 2 3,4, 9 16 5, ,
55
534 21 26
(1,2) 7, , 1
455 5 5
Maximum value of the direc
xy
xy
fxy xy yx
xy x
fxy fxy
yx yx
ff
f
Df
โˆ’โˆ’
=+
=โˆ’ =โˆ’+
โˆ’
โˆ’=โˆ’=โˆ’ โˆ’=โˆ’+=โˆ’
โˆ’
โˆ‡โˆ’ =โˆ’
=โˆ’โˆ’โˆ’โˆ’= โˆ’ = + = = โˆ’
โˆ’=โˆ’ โˆ’=โˆ’โˆ’=โˆ’
v
vvu
i
tional derivative at ( 1,2) is: ( 1,2)
25
49 25/ 4 49 16
Pf
โˆ’
โˆ‡โˆ’ =
+=+
2. Find the equation of the tangent line to the surface:
32
2 at (1,0,2)
yz
xyz e x P+=โˆ’ .
pf3
pf4

Partial preview of the text

Download Math 2224 Test 1: Calculus Problems and more Exams Calculus in PDF only on Docsity!

Math 2224 Test 1 Name__________Solution__________________ September 20, 2007 (pledged)

All work must be shown. Credit will not be given, even for correct final answers, unless all the correct and necessary calculations, procedures, and/or reasoning leading to such answers are shown. Calculators should not be needed and may not be used on this test. For numbers 1 - 6, please do all work on your own paper. Nothing on this page will be graded. You do not have to write the answers on this page. For number 7, matching, you should write the answers in the appropriate blank. When you are finished staple this page to the front of your work pages.

  1. Find the directional derivative for the function

2 ( , ) (^3) f x y x^ y y x

= + at the point P ( 1, 2) โˆ’ in the

direction of the point Q (2, โˆ’2). What is the maximum value of the directional derivative at P? 2 1 3 2 4 2 3

( 1, 2) 2 6 7, ( 1, 2)^1 1

( 1, 2) 7,^5

Maximum value of the direc

x y

x y

f x y x y yx x y x f x y f x y y x y x

f f

f

D f

= โˆ’^ + โˆ’

v โˆ’^ =^ โˆ’^ โˆ’^ = โˆ’^ โˆ’^ = โˆ’

v v u

i

tional derivative at ( 1, 2) is: ( 1, 2) 25 49 25 / 4 49 16

P โˆ’ โˆ‡ f โˆ’ =

  • = +
  1. Find the equation of the tangent line to the surface:

xyz^3 + e yz = 2 โˆ’ x^2 at P (1,0, 2).

3 2

3 3 2

The gradient vector of F at (1,0,2) is normal to the level surface containing the point (1,0,2) at P(1,0,2). ( , , ) 2 3 (1,0, 2) 2 10 0 Equation of t

yz

yz yz

F x y z xyz e x

F x y z yz x xz ze xyz ye F

i j k i j k angent plane:2( x โˆ’ 1) + 10( y โˆ’ 0) = 0 โ‡’ 2 x โˆ’ 2 + 10 y = 0

  1. Find the equation of the level curve for the function f ( , x y )= y + x^2 that contains the point

P (1,3)and sketch the level curve. Then sketch โˆ‡ f (3,1)with its initial point at (3,1). What is the

relationship between the gradient and the level curve. 2

2 2 2

2 1/ 2 2 1/ 2

2 2

f x y y x f y x y x y x

f x y y x x y x

f x y x y x y x

f

โˆ’ โˆ’

i j

i j

i j

-3 -2 -1 1 2 3

2

4

Note that the gradient vector (the short one!) is perpendicular to the tangent line to the level curve at (1,3).

  1. Find the linearization of f ( , x y ) = x^2^ + ln( x^2 + y )at the point P (2, โˆ’3). 2 2

2

2

( , ) ln( ), (2, 3) 4 ln1 4 0 4 2 4 ( , ) 2 , (2, 3) 4 8 1

( , ) 1 , (2, 3) 1

( , ) 4 8( 2) ( 3)

x x

y y

f x y x x y f x f x y x f x y

f x y f x y L x y x y

5 2.5^0 x-2.5-

-5 (^) -2.5 (^0) 2.5 (^5) y

0

2 z

5 2.5^0 x-2.5^ -

-4-5 (^) -2.5 (^0) y (^) 2.5 (^5)

0

2

4 z 4

1 0 x^ -

-1 (^0 ) y

-3.

-2. z

D E F

5 2.5^0 x-2.5^ -

-5 (^) -2.5 (^0) 2.5 (^5) y

0

2 z

1 0 x-

-1 (^0 ) y

0

1

z

1

1 0.5^0 x-0.5-

-1 (^0 ) y

-0.

0

z

G H

-7-7.5x^ -

-1 (^0) y (^1)

0

1 z 1

0 - 1 x -8 (^) -7.5y (^) -

0

1

z

0 - 1

1