MATH 115
SOME SAMPLE QUESTIONS FOR FINAL
1.Given f(x) = 6 x 2  4 , find and simplify
f ( x  h)  f ( x )
.
h
2.Which one is larger: log 3 8 or log 6 37 ? WHY?
3.Show that log a ( x  a 3  x )  log a ( x  a 3  x )  3
2( x  2)( x  3)
4.Consider the function f(x)=
x2
a. Find the x-intercept(s) of the graph of y=f(x)
b. Find the vertical asymptotes
c. Describe the end behavior
d. Sketch the graph of y=f(x). Label the intercepts and the asymptotes
2( x  2)( x  3)
e. Use the function above to find the domain of the function
x2
3x
5.Given the function f(x)=
5x  1
a. Find the inverse function, f 1 ( x)
b. Find f 1 (5)
6.Given f(x) = e x 5 and g(x)=ln(x), find
(a).(f og)(3)
(b).(g of) (4)
7. Evaluate each of the following:
(a) log 4 1 / 2 
(b) log 5 0.2 
(c) log 3
1

27

(d) 9
(e) log 4 2 
5
3
 t  2 , find sin t , tan t exact.
8. cos t  , and
8
2
log3 2
9.(a) Sketch the graph of g ( x)  ln( x  2)  3 below. Show all your steps. Label any
asymptotes and at least one point on your graph (put the coordinates of the point).
(b) Find the y-intercept of g(x).
(c) Find the x-intercept of g(x).
10. Consider the function f ( x)  5 cos(3x 

2
)
(a) What is the amplitude of f (x) ?
(b) What is the period of f (x) ?
(c) What is the phase shift of f (x) ?
(d) Graph of one period of f (x) on the set of coordinate axes below. Label the x-intercepts,
maximum and minimum values with their coordinates.
11.Find the domain of k(x) = log4 (x2 – x – 6)+
x2
. Express your answer using
( x  1)( x  5)
interval notation.
12. Solve each of the following equations (Express each answer as an exact values and give a
decimal approximation, accurate to two decimal places.)
(a) log 6 x  log 6 ( x  1)  log 6 20
(b) log 3 (log 4 x)  2
(c) 2 3 x 1  3 x  2
cos x
(The answer is 1 sin x )
sec x  tan x
tan x
 sec x  cos x
14.Are the following identities are true? a)
csc x
13. Simplify
b)
cot x. sec x
1
csc x
15.Given x = arcsin (.6) find the exact values of cos x, cos (2x).
3
7
16.Find the exact values of the expressions, tan(sin 1 ) and cos(tan 1 2) .
17.Given u=arcsin(0.7), find the following EXACT:
sinu, cosu, cos2u,sin(u/2)
18.There are two buildings. A man stands at a position where he can see both buildings. H e
knows that he is 300 feet from one building and 250 feet from the other building. If the angle
between the two lines of sight is 46 degrees, find the distance between the two buildings to the
nearest foot.
19. Solve for x: Find the EXACT solution(s) in the interval [0,2)
a. 2cosx.sinx-cosx=0
b. 3 2 sin x 1  1
20. Consider the function f(x)= 0.02 x 5  2 x  1
a. Sketch a graph of y=f(x) in the [-5,5] by [-5,5] window.
b. Give a numerical approximation of the zeros of f(x) accurate to one decimal place.
c. Use the information found in parts a and b above to find the solution to the inequality
0.02 x 5  2 x  1 <0 accurate to one decimal place.

 x  4, x  1


21.Given h(x)=  2,
1  x  3


 x, x  3

(a)
Find h(1) and h( 2) .
(b) Find
h(9)  h(9)
.
h(0)
22. When a certain drug is taken orally, the concentration of the drug in the patient’s
bloodstream after t minutes is given by C(t)= 0.06t  0.0002t 2 , where0  t  240 and the
concentration is measured in mg/L.
a. What is the concentration of the drug in the patient’s bloodstream after 10 minutes?
b. How many minutes is required to observe the concentration of the drug as 0.55 in the
patient’s bloodstream?
c. When the maximum serum concentration reached, and what is that maximum
concentration?
23. Problem 51 on page 440.
24. A navigator of a small plane is flying at an altitude of 4000 feet, and sees a small island
straight ahead of her. She determines that the angle of depression of the island shore closest to
her is 69 degrees and the angle if depression of the island shore farthest from her (along a
straight line) is 36 degrees. To the nearest foot, what is the distance between the two shores of
the island?