MATH 124-23:
TEST 1
EXTRA POINTS
FALL 2008
Directions: Show all work. An answer alone will not receive any credit!
(1) Let f be given by
f (x) = sin(x).
a) Find the equation of the tangent line to f at x = 0. Use the
following fact:
sin(x)
lim
= 1.
x→0
x
b) Find the equation of the tangent line to f at x = π2 .
c) At what point do the lines in a) and b) intersect?
d) Sketch f and the two lines above.
(2) Find the equation of the exponential function f which passes through
the points (1, 5) and (10, 2500). What is f (4)?
1
2
FALL 2008
(3) Graph the function g given by
g(x) = 32−x − 4.
To receive full credit, you must plot the x-intercept, the y-intercept,
one additional point, and indicate all vertical and (or) horizontal
asymptotes.
(4) Consider the function g given by
g(x) = 3 cos(2x + 1) − 5
a) What is the period of g?
b) What is the amplitude of g?
c) What is the maximum value g attains?
d) Sketch the graph of g over one full period.
(5) a) Write a sentence which describes the intermediate value theorem.
b) Consider the function g from the previous problem. How many
times does the function g take on the value −π over the interval you
sketched in part d) above. Justify your claim using the intermediate
value theorem.
MATH 124-23:
TEST 1
EXTRA POINTS
3
(6) Sketch the graph of
(x2 − 16)
.
x(x2 − 25)
To receive full credit, you must plot any x-intercepts, y-intercepts,
one additional point, and indicate all vertical and (or) horizontal
asymptotes.
y=
(7) Sketch the following functions:
f (x) = (x − 1)(x − 2)(x − 3)
g(x) = (x − 1)2 (x − 2)(x − 3)
h(x) = (x − 1)(x − 2)2 (x − 3)
i(x) = (x − 1)(x − 2)(x − 3)2
What effect does ”squaring a zero” have on the graph of the function
f?
4
FALL 2008
(8) Consider the function

x < 0,
 4x + a,
2
2(x − 1) , 0 ≤ x ≤ 2,
f (x) =

4x + a,
x > 2.
Here a is a constant.
a) For what value of a is f continuous at x = 0? Explain using a
limit calculation.
b) For the value of a found in a) above, is f differentiable at x = 0?
Explain using a limit calculation.
c) For what value of a is f continuous at x = 2? Explain using a
limit calculation.
d) For the value of a found in c) above, is f differentiable at x = 2?
Explain using a limit calculation.
MATH 124-23:
TEST 1
EXTRA POINTS
5
(9) A man kicks a soccer ball off the top of a building. The position of
the ball above the ground (in feet) t seconds later is given by

for 0 ≤ t ≤ 3,
 −10t2 + 10t + 60
−10t2 + 80t − 150
for 3 < t ≤ 5,
s(t) =

−10t2 + 110t − 300, for 5 < t ≤ 6.
a) Sketch the graph of this position function.
b) Describe the path of the ball.
c) How tall is the building?
d) What is the instantaneous velocity four seconds after the ball is
kicked?
(10) Using the limit definition, calculate the derivative of
x+3
f (x) =
x−4
at x = 5.