Download Fall 2011 Midterm 1 Exam for APPM 1350: Limits, Graphs, Continuity, and Derivatives and more Exams Calculus for Engineers in PDF only on Docsity!
APPM 1350 Midterm 1 Fall 2011
On the front of your bluebook, please write: a grading key, your name, student ID, section, and
instructor’s name (Chang, Curry, Dougherty, Guinn, Nelson). This exam is worth 100 points and
has 8 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.
- (16 points) Evaluate the following limits.
(a) lim t! 3
q
15
p 3 t
(b) lim x! 0
tan 7x
sin 2x
(c) lim v! 1
1 + v
|1 + v|
(d) lim x!
p x^2 + 4x
p x^2 + 9.
- (12 points) Match the graphs of the functions in Figure 1 to the graphs of their derivatives in
Figure 2. No explanation is necessary.
x
y
H 1 L
x
y
H 2 L
x
y
H 3 L
x
y
H 4 L
Figure 1: Functions
x
y
HaL
x
y
HbL
x
y
HcL
x
y
HdL
x
y
HeL
x
y
HfL
Figure 2: Derivatives
- (12 points)
f (x) =
2 x^2 if x 1
3 x if x > 1
(a) Use the definition of continuity to explain whether or not the function is continuous at x = 1.
(b) Use the definition of the derivative to explain whether or not the function has a derivative at x = 1.
- (11 points) Let f (x) = 3 +
2 x
(a) What is the domain of f (x)?
(b) Use the definition of the derivative to find f 0 (x) for all x in the domain of f.
(c) Find the equation of the tangent line to f at x = 1.
- (10 points) Show that the functions f (t) = 3t^2 and g(t) = cos t have an intersection point. Describe your reasoning and state any theorem you use.
- (12 points) Let g(x) =
p 5 x^2 + 2
2 x 5
. Use the appropriate limits to find all vertical and horizontal
asymptotes.
- (15 points) The figure at right shows the velocity^ v^ = s^0 (t) of a particle moving along a line, where s(t) is the position of the particle at time t seconds.
0.3 1 1.7 2 2.5 3
t
0
v
v á s ¢H t L
(a) Find the average acceleration of the particle over the interval 1 t 2. 5.
(b) Over what interval(s) is the particle moving in a positive direction?
(c) When does the particle reverse direction?
(d) Over what interval(s) is the acceleration of the particle negative?
(e) When does the particle move at greatest speed?
- (12 points)
(a) True or False: If the function f (x) is even, then the function g(x) = f (x) 1 is even.
(Use the definition of an even function to explain whether the statement is true or false.)
(b) True or False: The function g(x) =
p |x| has the same domain as the function h(x) = |
p x|. (Write down the domain of g and h to explain whether the statement is true or false.)
(c) Differentiate the function y = x^2
p 3 x^2
Extra Credit (5 points) Find the values of x that solve
6 +^
x
<^3.
