Answers to midterm review:
1. Write the equation of a line in point-slope form given: (-3, 2), m = 3/2
y – y1 = m (x – x1)
y – 2= 3/2(x + 3)
2. write the equation in slope-intercept form and standard form given: (1, 2), (-2, 8)
y = mx + b and Ax + By = C
m=
,
2 = -2(1) + b, 2 = -2 + b, 4 = b
y = -2x + 4
2x + y = 4
3.write the equation of a line that is parallel & perpendicular through a given point and a given line:
(-2, 4), y = 3x -2
4 = -6 + b
10 = b
y = 3x + 10
4 = 2/3 + b
10/3 = b
y = -1/3x + 10/3
4.Solve by factoring: 2x2 – 11x + 12 = 0
(2x – 3)(x – 4) = 0
2x – 3 = 0
x–4=0
x = 3/2
x=4
{3/2, 4}
5. Solve by quadratic formula: 3x2 + 4x = 10
3x2 + 4x – 10 = 0
=
6. Perform operations with complex numbers:
a. (2 – 3i) + (6 + 5i): 8 + 2i
b. (7 – 3i) – (6 – i): 1 – 2i
c. (7i – 3)(2 + 6i): -48 – 4i
d. 2 + 3i :
1 – 4i
7. Find the domain and range of the following functions:
a. f(x) =
:D: [-4, 4], R: [0, 4]
b. f(x) = x + 5: D: (- ∞, 8) (8, ∞), R: ( - ∞, 1) (1, (1, ∞)
x–8
8. determine if the following functions are bounded and how they are bounded. Then find their maximum and/or
minimum points
a. f(x) = -2x2 + 3 : bounded above, max: (0, 3)
b. f(x) = x3 – 9x: bounded, max: (-1.73, 10.39), min: (1.73, -10.39)
9. determine on what intervals the function is increasing and/or decreasing
a.
b.
Dec: [-6, 0], inc: [0, 6]
dec: (-∞, 1], inc: [1, ∞)
10. determine if the following functions are continuous or discontinuous. If they are discontinuous tell what type of
discontinuity is has (infinite, jump, or removable)
a. f(x) =
: continuous
b. f(x) = x + 5 : infinite discontinuity
x–8
11. determine the HA’s, VA’s, and EBA’s for the following functions
a. f(x) = 5x3
2x + 1
HA: none
VA: x = -1/2
EBA: y = 5/2x2 – 5/4x + 5/8
(EBA won’t be ugly on the test)
b. f(x) = 6x – 1
x3 – 1
HA: y = 0
VA: x = 1
EBA: y = 0
c. f(x) = 8x2 – 4x
2x2 – 4
HA: y = 4
VA: x =
EBA: y = 4
12. perform the following operations for the following functions:
f(x) = x + 5, g(x) = 2x2 – 4
a. f(x) + g(x): 2x2 + x + 1
b. f(x) – g(x): -2x2 – x + 9
c. f(x)g(x): 2x3 + 10x2 – 4x - 20
d. f(x)/g(x):
e. f(g(x)): 2x2 + 1
13. find the inverses of the following functions. Then determine if the inverse is a function.
a. f(x) = x2 – 16: f-1(x) =
, yes
b. f(x) = x + 2: omit
x
14. determine the transformations
a. f(x) = (x – 5)2 + 9: right 5, up 9
b. f(x) = -2|x| - 8: reflection over x-axis, vertical stretch of 2, down 8
c. f(x) = (x + 2)2: left 2
15. Find the vertex & axis of symmetry: f(x) = -3(x + 8)2 + 10: (-8, 10), x = -8 (axis of symmetry: x = x-coord. of vertex)
16. Find the vertex, axis of symmetry, and rewrite in vertex form: f(x) = 2x2 – 8x – 5: (2, -13), x = 2
17. Analyze the function: f(x) = 5 :
x2
to do: determine the domain, range, continuous/discontinuous, increasing/decreasing intervals, even/odd, boundedness,
local extrema (max/min), asymptotes, end behavior :
D: (-∞, 0) (0, ∞), R: (0, ∞), discontinuous, bounded below, no max/min, inc: (-∞, 0), dec: (0, ∞), even, VA: x = 0,
HA: y = 0,
18. Find the 0’s algebraically or graphically: f(x) = 5x3 – 5x2 – 10x: {-1. 0. 2} – graph on calculator and find the zeros
19. Find the cubic function given the zeros: 1, -2, -3: (x – 1)(x + 2)(x + 3), f(x) = x3 + 4x2 + x - 6
20. use long division to divide: f(x) = x3 – 3x + 4, d(x) = x + 2: x2 – 2x + 1 +
21. use synthetic division to divide: f(x) = 2x3 – 3x2 + 4x – 7, d(x) = x – 2: 2x2 + x + 6 +
22. Determine if k is an upper bound or a lower bound. (do synthetic division – if numbers are all positive it’s an upper
bound, if the number alternate between positive and negative it’s a lower bound)
a: f(x) = x3 – 3x2 + 4, k = -2: lower
b. f(x) = x4 – 2x3 + x2 - 9x + 2, k = 3: upper
a. -2
1 -3 0 4
-2 10 -20
1 -5 10 -16
b. 3 1 -2 1 -9 2
3 3 12 9
1 1 4 3 11
23. Using the remainder theorem find the remainders: f(x) = 2x4 - 9x3 + 7x2 - 5x + 11, x = 4
4 2 -9 7 -5 11
8 -4 12 28
2 -1 3 7 39
f(4) = 39
24. Using the Factor theorem find the factors of the following
a. f(x) = x3 + 3x2 – 13x – 15, x + 5: (x + 5)(x – 3)(x + 1)
a. -5
1 3 -13 -15
-5 10 15
1 -2 -3 0
x2 – 2x - 3
(x – 3)(x + 1)
a. 2
1 -5 2 8
2 -6 -8
1 -3 -4 0
x2 – 3x - 4
(x – 4)(x + 1) = 0
x = 4, x = -1
b. f(x) = x3 - 3x2 – 10x + 24, x – 2: (x – 2)(x – 4)(x + 3)
25. Using the factor theorem find the 0’s
a. f(x) = x3 - 5x2 + 2x + 8, x = 2: {-1, 2, 4}
b. f(x) = 4x3 - 12x2 - x + 3, x = 3: {-1/2, ½, 3}
26. Use the rational root theorem to find all real 0’s: f(x) = x3 - 7x2 + 2x + 40: {-2, 4, 5}
(find factors of leading coefficient and constant, then do
27. Use the rational root theorem to find all 0’s: f(x) = x3 - 5x2 + 4x – 20: {5,
}
, then do synthetic division
28. Solve the equation. Check for extraneous solutions
a. 1 - 2 = 4 : {1/4}
x x-2
b. x + 3 - 2 =
6
: {-1}
2
x
x+3
x + 3x
29. Graph: find all asymptotes and all intercepts
f(x) =
x+1
x2 – 3x – 10
EBA:_y = 0
VA(s):x = -2, x = 5
x-intercept(s): (-1, 0)
y-intercept: none
30. solve without a calculator:
a. log 1000 = x: 10x = 1000, 10x = 103, x = 3
b. log3
= x: 3x =
, 3x =
c. log5 x = 3: 53 = x, 125 = x
d. ln e4 = x: 4 = x
e.
= x: 4 = x
f.
= x: 8 = x
, 3x =
, 3x = 31, x = 1
31. expand or condense
a. log 8x: log 8 + log x
b. ln 4/x: ln 4 – ln x
c. log2 x3: 3
g. log 4 + log 6: log 4(6) = log 24
h. log x – log 4: log x/4
i. 4ln x + 6 ln y: ln x4/y6
32. given the logistic functions: f(x) =
find:
a. the HA’s: y = 0 and y = 1200
b. the y-intercept: (0, 12)
c. what is the initial population of deer at Briggs State Park? 12
d. In many years will it take to reach 500 deer? 10 years (this will happen sometime during the 10th year)
33. The population in a town in 2000 was 24,750. If the population is increasing at a rate of 2.5% per year find the
following:
a. Write the exponential function: P(t) = 24750(1.025)t
b. What would the population be in 2004? 27, 319
c. In what year will the population reach 40,000: 2019
34. There is 6.5 g of a certain substance has a half-life of 35 days.
a. Write the exponential function in terms of t.: A(t) = 6.5(
b. How much is left after 10 days: 5.33 g
c. When will there be less than 5 grams remaining? 14 days
35. Afghanistan suffered 2 major earthquakes in 1998. The one on February 4th had a Richter magnitude of 6.1, causing
2300 deaths and the one on May 30th measured 6.9 on the Richter scale, killing about 4700 people. How many times more
powerful was the deadlier earthquake?
log
= R1 & log
= R2
log
= 6.1 & log
= 6.9
(log
) - ( log
) = 6.9 – 6.1
log
- log
= .8
log
=.8 so 10.8 =
, 6.3, about 6 times greater
36. Solve the following equations. Round to the nearest thousandth
a.
b.
c.
d.
e.
2(1/4)4x = 32: (1/4)4x = 16, 4-4x = 16, 2-8x = 24, -8x = 4, x = -4/8, x = -1/2
6-2x = 18 : log6 6-2x = log6 18, -2x = log6 18, -2x = 1.61…, x = -.807
log4 x-4 = 4: 44 = x – 4, 256 = x – 4, 260 = x
4 + 3 ln (x – 3) = 8: ln (x – 3) = 4/3, eln x-3 = e4/3, x – 3 = 3.79…, x = 6.79
e-2x = 10: ln e-2x = ln 10, -2x = 2.30…, x = -1.15
37.You deposit $400 into an account that pays 3.5% simple interest. After 5 years, how much will you have?
A = 400(1 + .035)5, A = $475.07
38.You deposit $4500 into an account that pays 2.75% interest compounded quarterly. After 3 years, how much will you
have?
A = 4500 (1 +
, A = $4885.61
39.You deposit $1000 into an account that pays 2.4% interest compounded continuously, have much will you have after 8
years?
A = 1000e.024(8) = $1211.67
40.What is the interest rate compounded monthly is required for $500 to reach $800 in 5 years?
800 = 500(1 + 60, 1.6 = (1 + 60, 1.007… = (1 +
, .007… = r/12, .0943.. = r, 9.44%
41.Find the future value of a retirement account in which Zachary makes payments of $150 in which interest is credited
quarterly with an interest rate of 6.5% after 25 years.
FV = 150(
), $37,038.62
42.Find the present value of a loan in which payments of $247.60 are made quarterly with an interest rate of 3.25% for 6
years.
PV = 247.60(
, $5379.14
43.How much are the monthly payments on a loan in which $14,500 is borrowed at a rate of 2.25% for 3 years.
14,500(
= R(1 – (1 +
, $416.90
44.You contribute $50 per month into an account that earns 4.5% annual interest. After 30 years, how much will you have?
OMIT
45. convert from DMS to degrees: 49˚ 25’ 18”: 49 +
49.42
46. convert from degrees to radians: 60˚:
47. convert from radians to degrees:
48. evaluate on a calculator:
a. sin 32˚: .53
b. cos 18˚: .95
c. tan : -1.73
d. csc : 1.15
e. sec 50˚: 1.56
f, cot 25˚: 2.14
:
240
sin
49. Find all 6 trig ratios given:
, co
csc
50. given tan
= find the remaining 5 trig ratios:
sin
csc
3
8
51. Given the following triangle:
a. If
˚ and b = 12.5 find the remaining measurements
b. if
and c = 8.9 find the remaining measurements
a. Β = 52 , a: tan 38 =
c: cos 38 =
b.
c = 16
, a: cos 48 =
b: sin 48 =
= 5.96
, b = 6.61
52. point P (-3, 6) is on the terminal side of angle θ. Evaluate the 6 trig functions for θ. If the function is undefined, write
“undefined”
3
sin =
6
-3
53. evaluate without a calculator by using ratios in a reference triangle
a. tan 150˚:
b. sec
:
54. use quadrantal angles to find sin, cos, and tan. If the function is undefined, write “undefined”
a. . -270˚: sin -270 = 1, cos -270 = 0, tan -270 = ø
b.
: sin