MATH 1113
PRACTICE FINAL EXAM (I)
1. Find and simplify the difference quotient for the given function f ( x)  2 x  6 , that is, find
f ( x  h)  f ( x )
,h  0.
h
-----------------------------------------------------------------------------------------------------------------------------------------2. The graph of the function f ( x )  x is shown below. On the same xy- plane, use transformation rules to
graph g( x)  3 x  4  5 . Label three points on the graph of the transformed function.
(-2, 2)
(2, 2)
(0, 0)
-----------------------------------------------------------------------------------------------------------------------------------------3
3. Find the inverse function of f ( x)  8 x  1 .
-----------------------------------------------------------------------------------------------------------------------------------------4.
2
2
Given P ( x)   ( x  1) ( x  16) .
a. What is the degree of the function?
b. Determine the zeros of the function. State the multiplicity of each zero and if the
graph crosses the x-axis or touches the x-axis and turns around, at each zero.
Zero
Multiplicity
Odd/ Even
Behavior of the graph at the zero
a. Touches the x-axis and turns around
b. Crosses the x- axis
a. Touches the x-axis and turns around
b. Crosses the x- axis
a. Touches the x-axis and turns around
b. Crosses the x- axis
5. Radioactive Decay Radioactive iodine is used by doctors as a tracer in diagnosing certain thyroid gland
disorders. This type of iodine decays in such a way that the mass remaining after t days is given by the
function 𝑚(𝑡) = 6.1𝑒 −0.087𝑡 , where 𝑚(𝑡) is measured in milligrams.
a. The doctor administers the tracer dose on Day Zero. Find the amount at time 𝑡 = 0.
b. How long will it take for the radioactive iodine to decay to 10% of the original amount. Round the answer to
the nearest whole number.
-----------------------------------------------------------------------------------------------------------------------------------------x2
6. Given the rational function r ( x ) 
,
( x  3) ( x  5)
a. Find the vertical asymptotes of the graph of r (x ) , if any.
b. Find the x intercepts of the graph of r (x ) .
c. Find the y intercept of the graph of r (x ) .
d. Find the horizontal asymptote of the graph of r (x ) , if there is one.
e. Use parts (a – d) to sketch the graph of r (x ) . Label the vertical and horizontal asymptotes.
-----------------------------------------------------------------------------------------------------------------------------------------7. Use the properties of logarithms to expand the logarithmic expression as much as possible.
 x3 y 

ln 
 ( x  1) 9 


-----------------------------------------------------------------------------------------------------------------------------------------8. Simply each expression without using a calculator.
a. 3
log 3 
b. log 2 2
x
-----------------------------------------------------------------------------------------------------------------------------------------9. Solve the logarithmic equation log 2 (𝑥 + 6) − log 2 (𝑥) = 2
10. Answer.
20𝜋
10.a) The angle 𝜃 = 3 in standard position.
10.b) State an angle between 0 and 2 𝜋 which is
coterminal to the angle 𝜃.
-----------------------------------------------------------------------------------------------------------------------------------------11. Given the triangle below, find the exact value of csc 𝜃.
C
a=25
b
𝜃
A
c=24
B
-----------------------------------------------------------------------------------------------------------------------------------------17𝜋
12. Find the exact value of tan (− 6 ).
-----------------------------------------------------------------------------------------------------------------------------------------13. If a distance from a helicopter to a landing pad is 300 feet and the angle of depression is 40 . Rounded to
the nearest foot, find the distance on the ground from a point directly below the helicopter to the landing
pad.
-----------------------------------------------------------------------------------------------------------------------------------------14. Use an addition or subtraction formula to find the exact value of cos 15º.
-----------------------------------------------------------------------------------------------------------------------------------------15. Find the exact values of the following expressions:
1
a. tan (cos−1 (− 4))
7𝜋
b. sin−1 (sin (
4
))

1
16. Given the trigonometric function f ( x )  2 cos  x  
2
4
a. Determine the amplitude of f ( x ) .
b. Determine the period of f ( x ) .
c. Determine the phase shift of f ( x ) .
d. Sketch one period of the graph of f ( x ) .Label the x and y intercepts.
---------------------------------------------------------------------------------------------------------------------------------17. If tan(t ) 
 12
, and t is an angle in quadrant IV, find the exact value of the following
9
a. tan(2t )
t
b. cos 
2
---------------------------------------------------------------------------------------------------------------------------------18. Verify the trigonometric identity
csc t  cos t cot t  sin t
-----------------------------------------------------------------------------------------------------------------------------------------19. Find all the exact solutions to the following equation on the interval [0,2 ) .
sin x  2 sin x cos x  0
-----------------------------------------------------------------------------------------------------------------------------------------20. Refer to the figure below, depicting a triangle with sides and angles as shown.
Use the Law of Sines and/or the Law of Cosines to find the
following:
a. The length of the side b to the nearest tenth.
b. The measure of the angle  A in degrees
rounded to the nearest whole number.
Addition and Subtraction Formulas:
sin(u  v)  sin u cos v  cos u sin v
sin(u  v)  sin u cos v  cos u sin v
cos(u  v)  cos u cos v  sin u sin v
cos(u  v)  cos u cos v  sin u sin v
tan u  tan v
1  tan u tan v
tan u  tan v
tan(u  v ) 
1  tan u tan v
tan(u  v ) 
Double Angle Formulas:
sin(2u)  2 sin u cos u
2
2
cos(2u)  cos u  sin u
2
cos(2u)  2 cos u  1
2
cos(2u)  1  2 sin u
2 tan u
tan(2u) 
2
1  tan u
Half-Angle Formulas:
sin
u
1  cos u

, (± 𝑑𝑒𝑝𝑒𝑛𝑑𝑠 𝑜𝑛 𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡)
2
2
u
1  cos u

, (± 𝑑𝑒𝑝𝑒𝑛𝑑𝑠 𝑜𝑛 𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡)
2
2
u 1  cos u
sin u
tan 

2
sin u
1  cos u
cos
Law of Sines:
a
b
c


, where a, b, c are lengths of sides and A, B, C are the opposite angles.
sin A sin B sin C
Law of Cosines:
2
2
2
a  b  c  2bc cos A , where a, b, c are lengths of sides and A is the angle opposite sid
2
2
2
b  a  c  2ac cos B , where a, b, c are lengths of sides and B is the angle opposite side b.
2
2
2
c  a  b  2ab cos C , where a, b, c are lengths of sides and C the angle opposite side c.
Arc Length formulas:
s  r , where  is a central angle.