M161, Survey of Calculus. Exam 1. Feb 2010 Name: _____
_________________
NO CALCULATORS. FOR OPTIMAL CREDIT, SHOW WORK (WHERE APPLICABLE)
1)
(23 pts) For parts a-d, determine the following limits, if they exist
a.)
b.)
c.)
2)
2x 2 + x − 3
lim
x →1
3x 2 − 3
3
lim 15 − 3x
x →∞
2x 3 + 1
lim
x → −3 x2
+ 2x + 3
d.)
3x − 1

Given f(x) = 4
2x

e)
For the piece-wise function in part d.) above, f(x) is continuous. True or False
if
x < 1
lim

x = 1 Find x →1 f(x)
x > 1
’
(12 pts) Use the limit definition of the derivative to find f (1) when f(x) = - 2x2 + 3x
3)
(25 pts) Differentiate the following functions. There is no need to simplify!!!
(5 x − 8) 5
a)
f(x) =
b)
f(x) =
c)
f(x) = ( x3 – 3x +5 ) (4x5 – 3x4 +
d)
f(x) = 2 ( 7x+1 ) 15
e)
 2  πx
f(x) =   +
3
5
4
3x +
3
9
x
.
2
)
x
4)
(20 pts) Given for parts a-d:
f(x) =
x
x +1
2
a)
Find the derivative of f(x)
b)
Determine the slope of the tangent line of f at the point ( 2,
c)
Give the equation of the tangent line at the point ( 2,
d)
Find the x-value(s) on the graph of f where the tangent line is horizontal.
2
)
5
2
)
5
3+
xy = 5y2
5)
(10 pts) Implicitly differentiate the following functions:
6)
(10 pts) The volume of a very large spherical balloon with radius r is changing with respect to
time. At a certain instant of time, the radius of the balloon is 10 meters and its radius is
decreasing at the rate of 0.2 meters per hour. How fast is the volume decreasing at that instant
of time?