Pre-Calc Finals
Find lim n->-infiity(1.28)^n
0
Solve algebraically csc x = 6 for 0 <_ x <_ 2pi. Show work and give answers to 3 decimal places.
0.167
Rewrite the statement ln(1/8) = -2.079 as an exponent.
1/8 = e^-2.079
Graph to find all the solutions to sin theta = 0.2 on the interval 3pi <_ theta <_ 5pi
12.768, 15.507
Determine the amplitude of the periodic function y = -4cos(2x) - 2
4
Formula for Compounded Yield.
A = P(1+r/n)^nt
Use the triangle from #11. If A = 24 degrees, B = 66 degrees and c = 7. Find values for a and b. Express your answer to two decimal places.
a = 2.85 b = 6.39
Find A and c if a = 9, b = 14 and B = 11 degrees. Round to two decimal places. A is the top corner, B is the right corner, C is the left corner. A is next to c, b is on top of C, and a is to the left of B.
a = 7.05 degrees c = 22.73
Suppose you would like to have $20,000 in 10 years. What is the minimal amount you need to deposit into a bank account earning 3% compounded monthly to reach this goal?
$14,821.40
Kathleen opens a savings account with $1400. The account earns 3.3% interest compounded weekly. How much will be in the account after 13 years?
$2149.72
Find the point on a circle with radius 4.6 determined by the angle 200 degrees. Round to three decimal places.
(-4.323,-1.573)
Simplify 2sin(t) + 6(1 - 2sin(t))
-10sint + 6
Calculate the average rate of change for f(x) = -x^2+3x between x=2 and x=3.
-2
One solution to the equation 5 + x = 5(2)^x is 0. Use your calculator to find the other solution to 2 decimal places.
-4.82
Determine the midline of the function y = 4cos(4x) - 5
-5
If f(x) (1/(x+4)) + 1 solve f(x) = 0 for x. Round to 2 decimal places.
-5
How many solutions to sin x = -1/2 are there 0 <_ x <_ pi?
0
Solve algebraically tan x = 2 for 0 <_ x <_ 2pi. Show work and give answer to 3 decimal places.
1.373, 4.515
The hydrogen concentration [H+] for a certain substance is 0.025119 moles/liter. What is the pH to 1 decimal place?
1.6
To measure the height of a mountain, a person stands a certain distance from the base and measures the angle of elevation to the top of the mountain. This turns out to be 70 degrees. Moving 100 feet closer, the angle of elevation to the top of the mountain is 73 degrees. How tall is the mountain? Round to 2 decimal places.
1717.05 feet
Suppose you are on a Ferris wheel that turns in a counter clockwise direction. And your height, in meters, above the ground at time t, is given by h(t) = 17sin(pi/2 t) + 19. How many meters above the groundd are you at time t=0?
19 meters
Simplify 2sin^27t + 5cos^27t. Express your answer in terms of cos.
2 + 3cos^27t
A ramp is 101 feet long and the change in height of the ramp is 5 feet. What angle does the ramp form with the horizontal?
2.838 degrees
The minute hand on a watch is 0.8 inches long. How many inches does the tip of the minutes hand travel as the hand turns through 210 degrees? Round to two decimal places.
2.94 inches
What is the half-life of a substance that decays at a rate of 3.25% per day? Round your answer to the nearest hundredth of a day.
20.98 days
In the following figure, the coordinates of P are (0.48, 0.87). The angle theta = _______ degrees. Round to the nearest whole number.
241 degrees
Suppose there are 110 people living in a small town and the rate of change of the town's population is 9 people per year. How many people will be living in the town twenty years from now?
290
If 0 <_ t <_ pi, in what quadrant is cos t = -0.18?
2nd quadrant
What is the hydrogen ion concentration [H+] for a substance with a pH of 5.5? Expand the power of ten to 3 decimal places.
3.162 * 10^5
Solve the equation for the value of theta. 6costheta - 2 = 3 Give the answer both in radians and degrees.
33.557 degrees 0.586 radians
The U.S. population 2005 was approximately 296.4 million. Assume the population increases at a rate of 1.24% per year. How many million people would you expect to live in the U.S. in 2020?
356.6 million
If 11% of a radioactive substance decays continuously in 7 hours. What is the half-life of the substance in hours? Round to 3 decimal places.
40.773 hours
The price of an item increases due to inflation. Let p(t) = 87.50(1.057)^t. At what continuous rate is the price increasing? Round the percent to 2 decimal places.
5.54%
An ant walks up a ramp at an incline of 13 degrees. If after 2 minutes his vertical distance from the ground is 13 inches, how far has the ant traveled in the horizontal direction?
56.31 inches
What is the horizontal shift of y = 3 cos (8t + 48) - 2?
6 to the left
For the continuous compound interest in the formula Q = 2.500e^0.06t find the effective annual growth rate. Round to the nearest hundredth of a percent.
6.18%
Use the figure below to find the missing side c, when a = 10, b = 7 and C = 43 degrees. Round to 2 decimal places. A is the top corner, B is the right corner, C is the left corner. A is next to c, b is on top of C, and a is to the left of B.
6.83
Rewrite e^6a = b using logs.
6a = ln b
Find the value of the expression x^2-3xy if x=3 and y=1/5.
7.2
Write (7 cos (x-4)) / (4 sin (x-4)) in terms of the cotangent function.
7/4 cot (x-4)
A rope is attached to the top of a tree and to a stake in the ground. The angle of the rope with the ground is 35 degrees and the rope comes out 140 feet from the tree. How tall is the tree?
98.03 feet
A vacationer sits all day on the corner of a pier in Boston Harbor and notices that at 9 a.m., when the water level is at its lowest, the water's depth is 2 feet. At 4 pm, the water has risen to its maximum depth of 12 feet. If the depth of the water level varies periodically, let f(t) be the formula for the depth of the water, in feet, as a function of time t, in hours past 9 am. Determine the amplitude, midline and period and B for the above scenario.
A = 5 ft K = 7ft P = 14 hours B = pi/7
Find the measures of the angles of the triangle is a = 7, b = 8, and c = 4. Round to two decimal places.
A = 61.03 degrees B = 88.98 degrees C = 29.99 degrees
Which bank has the best effective annual yield? Bank 1 with a nominal rate of 6.4% compounded monthly Bank 2 with a nominal rate of 6.33% compounded weekly Bank 3 with a nominal rate of 6.55% compounded yearly
Bank 1
[1 + (i/n)]n - 1
Formula for Effective Yield.
State the midline and amplitude of the function y = (11 - 3sint)/44
K = 1/4 A = 3/44
If you were to draw a graph representing the total population, P, of a city (t) years after 1900, which axis would t be on? a. Horizontal b. Vertical
a
Which of the following quotients is equivalent to sec t? a) 1/cost b) 1/sint c) 1/tant
a
Given f(x) = 4x^2 - 2 and g(x) = -3x + 1, find a) f(g(8)) b) g(f(8))
a) 2114 b) -761
The populations of 4 species of animals are given by the following equations: P1 = 290(0.9)^t P2 = 890(1.22)^t P3 = 520(0.8)^t P4 = 560(1.05)^t a) Which species are growing in size? b) What is the annual percent growth rate for the population that is shrinking the fastest? c) What is the largest initial population of the 4 species?
a) P2 and P4 b) -20% c) P2 @ 890
The following chart gives the number of students in a class that are a specific height in inches. 55 inches - 4 60 inches - 7 65 inches - 4 70 inches - 2 75 inches - 1 a) Is the number of students in each category a function of the height? b) Is the height in each category a function of the number of students in that category?
a) Yes b) No
Find the following exactly using the figure given if a = 6 and b = 5. Express your answer as an unsimplified radical. a) sin B b) cos B c) tan B
a) sin B = square root of (36+25) b) cos B = 6/c c) tan B = 5/6
If f(x) is an increasing function, what can you say about f(-4) and f(3)? a. F(-4) > f(3) b. f(-4) < f(3) c. f(-4) = f(3) d. It cannot be determined.
b
Find side b when A = 54 degrees, B = 51 degrees and c = 5. Round to two decimal places. A is the top corner, B is the right corner, C is the left corner. A is next to c, b is on top of C, and a is to the left of B.
b = 4.02
What is the domain of the function f(x) = 6/square root of (9-x^2) a. -6 < x < 6 b. -6 <_ x <_ 6 c. -3 < x < 3 d. -3 <_ x <_ 3
c
Simplify: (cos2theta + 1)/cos theta
cos theta
Prove the identity: show work. 1 + sin theta / cos theta = cos theta / 1 - sin theta
cos^2 theta = 1 - sin^2 theta
For f(x) = x^2 - 4x + 5, what is f(x+4)? a. x^2 + 12x -37 b. x^2 - 12x - 37 c. x^2 - 4x + 5 d. x^2 + 4x + 5
d
Let w(m) give the weight (in pounds) of an average-sized baby girl who is m months old. What does it mean if w(4) = 13? a. 13 baby girls weigh 4 pounds b. 4 baby girls weigh 4 pounds c. an average 13-month old girl weighs 4 pounds d. an average 4-month old girl weighs 13 pounds
d
Find the domain and range of h(x) = -7/(x^2+4)
d: all real numbers r: -7/4 <_ x <_ 0
What is the range of the function y = 6/square root of (4-x^2) a. y _> 2 b. y <_ 2 c. y _> 6 d. y <_ 6 e. y _> 3 f. y <_ 3
e
A vacationer sits all day on the corner of a pier in Boston Harbor and notices that at 9 a.m., when the water level is at its lowest, the water's depth is 2 feet. At 4 pm, the water has risen to its maximum depth of 12 feet. If the depth of the water level varies periodically, let f(t) be the formula for the depth of the water, in feet, as a function of time t, in hours past 9 am. Write the function.
f(t) = -5cos(pi/7 t) + 7
If y = f(x) = x^3 + 6, what is the inverse function f^-1(y)?
f^-1(y) = cubed root of (y-6)
A box with a volume of 150 ft^3 has a square base of side length s ft. and a height of h ft. Write a formula for the height of the box as a function of the side length of the box.
h(s) = 150/s^2
Each of the following functions in the table below is increasing, but each increases in a different way. Determine which is linear, which is exponential and which is neither. (Assume it's in increments of 1) f(t) 14.46 18.36 22.26 26.16 30.06 33.96 g(t) 9 19 28 36 43 49 h(t) 52.65 61.60 72.07 84.32 98.66 115.43
linear: h(t) exponential: f(t) neither: g(t)
What is the period of y = 3sin(4t-2) + 5
pi/2
Find algebraically all the solutions to -3 cos t - sin t cos t = 0 for 0 <_ t <_ 2pi. Express with pi in the solutions.
pi/2, 3pi/2
Simplify the expression to contain only sin t, cos t, and tan t. cos t tan t sin t
sin t
The sec theta = 17. Find the sin theta and the tan theta.
sin theta = 0.998 tan theta = 0.042
Solve: 800e^0.7t = 400e^0.5t Round to 3 decimal places.
t = -3.466
Solve for t: log (t-375) = 1
t = 385
Let f(x) = 4/(square root (x+45)) and g(x) = x^2 - 14x. Find the domain of f(g(x)).
x > 9