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Infinite Limits - Calculus I - Exam, Exams of Calculus

Anna UniversityCalculus

Infinite Limits, Horizontal and Vertical Asymptotes, Definition of Derivative, Sketch Graph, Slope of Line Tangent, Equation of Line Tangent, Average Velocity, Continuous Function are some important points from this exam paper of Calculus I.

Typology: Exams

2012/2013

Uploaded on 03/15/2013

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Test 1
January 24, 2003
Calculus I Name
You may not use calculators, notes, or books. Do your own work.
Justify your answers mathematically. `Show your work.' CIRCLE ANSWERS.
A.In each of the following, nd the limit, if it exists. Innite limits are allowed.
If a limit fails to exist, so state [4 each].
1. lim
x!2(x23x+ 2)
2. lim
y!2
y27y+10
y2
3. lim
t!3
t3
t26t+9
4. lim
x!1
35x+5x2
7x2+4x+2
5. lim
x!3
pxp3
x3
6. lim
h!0
(2+h)38
h
7. lim
x!5+
1x
x5
8. lim
x!3
x29
jx3j
9. Find all horizontal and vertical asymptotes and sketch the graph.of y=x+3
x2:
10. Use the denition of derivative to nd f0(a) for f(x) = x3:
11. Use the denition of derivative to nd f0(a) for f(x) = 1
x:
12. Let g(x) = px. (a) Find the slope of the line tangent to the graph of y=g(x) at
x= 9. (b) Find the equation of the line tangent to the graph of y=g(x) at the point
(9;3):
13. Suppose a particle moves along a line and its position at time tis given by s(t) = 3t2t2:
Find (a) the average velocity from t= 1 to t= 3:(b) Use limits to nd the instanta-
neous velocity at t= 3:
14. Sketch the graph of a continuous function y=g(x) whose derivative g0(a) has the
following properties: g0(a)>0 for all a < 3; g0(a)<0 for 3 <a<6;and g0(a)>0 for
a > 6:
15. f(x) = x
x21
pf2

Partial preview of the text

Download Infinite Limits - Calculus I - Exam and more Exams Calculus in PDF only on Docsity!

Test 1

January 24, 2003

Calculus I Name You may not use calculators, notes, or books. Do your own work. Justify your answers mathematically. `Show your work.' CIRCLE ANSWERS.

A.In each of the following, nd the limit, if it exists. In nite limits are allowed. If a limit fails to exist, so state [4 each].

  1. lim x! 2 (x^2 3 x + 2)
  2. lim y! 2

y^2 7 y+ y 2

  1. lim t! 3

t 3 t^2 6 t+

  1. (^) xlim!1^3 ^5 x+5x

2 7 x^2 +4x+

  1. lim x! 3

pxp 3 x 3

  1. lim h! 0

(2+h)^3 8 h

  1. lim x! 5 +

1 x x 5

  1. lim x! 3

x^2 9 jx 3 j

  1. Find all horizontal and vertical asymptotes and sketch the graph.of y = x x+3 2 :
  2. Use the de nition of derivative to nd f 0 (a) for f (x) = x^3 :
  3. Use the de nition of derivative to nd f 0 (a) for f (x) = (^) x^1 :
  4. Let g(x) =

p x. (a) Find the slope of the line tangent to the graph of y = g(x) at x = 9. (b) Find the equation of the line tangent to the graph of y = g(x) at the point (9; 3):

  1. Suppose a particle moves along a line and its position at time t is given by s(t) = 3t 2 t^2 : Find (a) the average velocity from t = 1 to t = 3: (b) Use limits to nd the instanta- neous velocity at t = 3:
  2. Sketch the graph of a continuous function y = g(x) whose derivative g^0 (a) has the following properties: g^0 (a) > 0 for all a < 3 ; g^0 (a) < 0 for 3 < a < 6 ; and g^0 (a) > 0 for a > 6 :
  3. f (x) = (^) x 2 x 1

D.[6] Use the di erence quotient and limit to nd the following:

f 0 (3) if f (x) = x^2.

E.[6 each]Use di erentiation rules to nd f 0 (x); and simplify answers, if:

  1. f (x) = 3x^7 + 5x^5

p 3 x + 12

  1. f (x) = 53 xx+3 1
  2. f (x) = (^1) x + x 3
  3. f (x) = x

3 1 x^4

F.[6 each] Work the following:

  1. Suppose a; b; c are constants and N is a positive integer. If z = ayN^ + by + c, nd dzdy :
  2. Let y = 2x^4 5 x^3 + x^2 8 x +

p 32 : 78 : Find y^00 = d

(^2) y dx^2 :

  1. Find the equation of the line tangent to y = x 2 x^3 at (x; y) = (1; 1).
  2. Find all points on the curve y = x^3 12 x + 10 where the tangent is parallel to the x-axis.

G. [10] Suppose a body moves upward in a vertical line under gravity alone (air resis- tance, etc., neglected) with initial velocity of 64 ft/sec. Using the calculus methods as discussed in class, nd:

(a.) An expression for its height s (in ft) at time t;

(b.) the length of time it rises;

(c.) how high it goes.