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Spring 2012 Final Exam in Calculus: Mathematical Functions and Integration, Exams of Calculus for Engineers

Damodaram Sanjivayya National Law UniversityCalculus for Engineers

The final exam for a calculus course during the spring 2012 semester. The exam covers various topics related to mathematical functions, limits, derivatives, integrals, and continuity. Students are required to evaluate expressions, perform integrations, match functions to their graphs, and find derivatives. No calculators or electronic devices are allowed.

Typology: Exams

2012/2013

Uploaded on 02/25/2013

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APPM 1345/1350 Final Exam Spring 2012
On the front of your bluebook, please write: a grading key, your name, student ID, section, and instructor’s
name (Chang or Guinn). This exam is worth 150 points and has 13 questions. Show all work! Answers with no
justification will receive no points. Please begin each problem on a new page. No notes, calculators, or electronic
devices are permitted.
1. (8 points) Match the following functions to their graphs in Figure 1. No explanation is necessary.
(a) y=ex−1(b) y= ln(−x)(c) y= sin−1x(d) y= cosh x.
x
y
H1L
x
y
H2L
x
y
H3L
x
y
H4L
x
y
H5L
x
y
H6L
x
y
H7L
x
y
H8L
Figure 1: Functions
2. (18 points) Evaluate the following expressions.
(a) lim
x→∞
e2x+e−2x
e2x−e−2x(b) lim
t→0+(cos t)2/t2(c) d
dx Z2
2/xpt3−9dt
3. (12 points) Evaluate the following integrals. Simplify your answers.
(a) Zx
(2x−1)2dx (b) Ze
1
log2x
xdx
4. (10 points) Let
f(x) =
ex2+ 1
4
πtan−1x+ 1, x 6= 1
e+ 1, x = 1.
Is fcontinuous at x= 1? Justify your answer.
5. (8 points) Find dy/dx by implicit differentiation. Leave your answer unsimplified.
exy + cos x= ln y
pf3

Partial preview of the text

Download Spring 2012 Final Exam in Calculus: Mathematical Functions and Integration and more Exams Calculus for Engineers in PDF only on Docsity!

APPM 1345/1350 Final Exam Spring 2012

On the front of your bluebook, please write: a grading key, your name, student ID, section, and instructor’s name (Chang or Guinn). This exam is worth 150 points and has 13 questions. Show all work! Answers with no justification will receive no points. Please begin each problem on a new page. No notes, calculators, or electronic devices are permitted.

  1. (8 points) Match the following functions to their graphs in Figure 1. No explanation is necessary.

(a) y = ex−^1 (b) y = ln(−x) (c) y = sin−^1 x (d) y = cosh x.

x

y

H 1 L

x

y

H 2 L

x

y

H 3 L

x

y

H 4 L

x

y

H 5 L

x

y

H 6 L

x

y

H 7 L

x

y

H 8 L

Figure 1: Functions

  1. (18 points) Evaluate the following expressions.

(a) lim x→∞

e^2 x^ + e−^2 x e^2 x^ − e−^2 x^

(b) lim t→ 0 +

(cos t)^2 /t

2 (c)

d dx

2 /x

t^3 − 9 dt

  1. (12 points) Evaluate the following integrals. Simplify your answers.

(a)

x (2x − 1)^2

dx (b)

∫ (^) e

1

log 2 x x

dx

  1. (10 points) Let

f (x) =

ex

2

  • 1 4 π tan

− (^1) x + 1 ,^ x^6 = 1

e + 1, x = 1. Is f continuous at x = 1? Justify your answer.

  1. (8 points) Find dy/dx by implicit differentiation. Leave your answer unsimplified.

exy^ + cos x = ln y

  1. (12 points) Match the graphs of the functions in Figure 2 to the graphs of their derivatives in Figure 3. No explanation is necessary.

x

y

HaL

x

y

HbL

x

y

HcL

x

y

HdL

Figure 2: Functions

x

y

H 1 L

x

y

H 2 L

x

y

H 3 L

x

y

H 4 L

x

y

H 5 L

x

y

H 6 L

x

y

H 7 L

x

y

H 8 L

Figure 3: Derivatives

  1. (12 points) Sketch a graph of a single function y = g(x) that satisfies all of the following conditions. No explanation is necessary.

(a) lim x→∞ g(x) = − 2 (c) lim x→ 1 g(x) = 2 (e) g′′(6) < 0

(b) lim x→ 0 +

g(x) = −∞ (d) lim h→ 0

g(1 + h) − g(1) h = 2 (f)

1

g(x) dx < 0

  1. (10 points) Consider the functions

f (x) = arcsin(ln x), g(x) = ln(arcsin(x)).

(a) Find the domains of f (x) and g(x). (b) Which is larger: f (1) or g

2

  1. (12 points) Farmer Joe needs to fence off two sections of pas- ture next to the river: one for his goats, and one for his horse. He needs each section of pasture to be 60 , 000 (for a total of 120 , 000 ) square feet and wants each to have the same dimen- sions. If he does not need any fence along the river, what di- mensions should he make the pasture to minimize the amount of fencing he needs to use? Draw a diagram and clearly label the optimal dimensions.