calculus 1
Calculus I
Directions: (10 pts. each) Answer each of the following questions below. In order to receive ANY credit for a question, you must SHOW YOUR WORK using proper notation and clear and concise logic. You're graded on both the accuracy of your answers AND your explanations that sufficiently support your answers. Unless otherwise stated, you're to give the EXAXCT VALUES of answers instead of decimal approximations. In order to receive ANY credit for any applied/word problem (i.e. Problems #29 - ), you MUST declare a variable (unless the variable(s) have already been declared in the problem) and set up and solve an appropriate mathematical expression that can be used to answer the question. Proper units must also be included in answers to applied problems. NO CREDIT WILL BE GIVEN FOR EITHER GUESSING OR CHECKING POSSIBLE ANSWERS WITHOUT SOLVING THE PROBLEM. YOU CANNOT USE CALCULUS TO SOLVE THESE PROBLEMS.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Multiply or divide as indicated. Write your answer in factored form.
1) x22 - 9x + 14 · xx22 -- 1618x x ++ 4877 1)
2)
x
-
12
x
+
32
Simplify the complex rational expression.
4
x
2
-
4
x
-
32
-
1
x
-
8
2)
1 + 1 x + 4
Find the difference quotient for the function and simplify it.
3) g(x) = 6x2 + 14x - 1 3)
Find the domain and range of the function. Write your answers using interval notation.
4)
g(z)
=
16
-
z
2
4)
Find a formula for the function graphed.
5) 5)
Determine if the function is even, odd, or neither. You must use algebra to justify your answer; otherwise, no full credit will be given. NO CREDIT is given for an answer without a mathematical explanation.
6) f(x) = x -+7 9 6)
State the domain of the composition.
7)
(
g
H
h)(x) with g(x)
=
x
+
5
and h(x)
=
8
x
+
7
7)
Compute
f(x
+
h)
-
f(x)
h
(h
J
0) for the given function
.
8) f(x) = 4x - 8 8)
9)
f(x)
=
5
x
2
+
6
x
9)
10)
f(x)
=
1
9
x
10)
Solve the equation by multiplying both sides by the LCD.
11) 32x - x 3+ 1 = 1 11)
12)
Solve the equation.
x
+
6
+
2
-
x
=
4
12)
13)
(
4
x
-
2
)
/
3
2
+
6
=
15
13)
14)
3
x
+
4
=
x
-
1
14)
Find the real solutions of the equation by factoring.
15) x3 + 8x2 - x - 8 = 0 15)
Solve the equation by making an appropriate substitution.
16) (x2 - 2x)2 - 11(x2 - 2x) + 24 = 0 16)
Solve the logarithmic equation.
17) log2(x + 7) + log2(x - 7) = 2 17)
Solve the exponential equation. Express the solution set in terms of natural logarithms.
18) 4x + 4 = 52x + 5 18)
Solve the inequality and express the solution in interval notation.
19) 7Ax - 1A L 2 19)
Solve the inequality. Write your answer using interval notation.
20) x 18- 5 > x 15+ 1 20)
Write the equation as f(x) = a(x - h)2 + k. Identify the vertex, range, and axis of symmetry of the function.
21) f(x) = x2 + 5x + 2 21)
23) log
Find a and k and then evaluate the function. Round your answer to three decimal places when necessary.
22)
f(x)
=
a
2
+
8
e
kx
with f(0)
=
-
2
and f(2)
=
-
4
. Find f
(1).
22)
Use the change
-
of
-
base formula and a calculator to evaluate each logarithm. Round your answer to three decimal places.
2
142.6
23)
Use the fundamental identities to find the value of the trigonometric function.
24) Find sin ΅, given that sec ΅= and ΅ is in quadrant IV. 24)
7
45
45
Use the given information to find the exact value of the expression.
25) Find cos (΅- Ά). sin ΅= , ΅ lies in quadrant II, and cos Ά = , Ά lies in quadrant I. 25)
4
5
2
5
Use the given information to find the exact value of the function .
26)
cos (2
΅
)
=
1
9
and
3
Δ
2
K
2
΅
<
2
Δ
Find sin
΅
.
26)
Use the given information given
to find the exact value of the trigonometric function.
27)
sec
Ό
=
-
5
3
,
Ό
lies in quadrant II
Find sin
Ό
2
.
27)
Find all solutions to the equation. Express each result in radians.
28) 2 sin2 x + sin x = 1 28)
Find all real numbers in the interval [0, 2Δ) that satisfy the equation.
29) 4 sin2x = 4 cos x + 1 29)
30) 2
3
sin 4x
=
3
30)
31)
sin x
=
2
cos
x
2
31)
Solve the problem.
32) Find the exponential function of the form f(x) = aebx that passes through the points 32)
(0
,
3
)
and
(
3
,
9
)
.
33) The cost of owning a home includes both fixed costs and variable utility costs. Assume that 33) it costs $4450 per month for mortgage and insurance payments and it costs an average of $2.16 per unit for natural gas, electricity, and water usage. (i) Determine a linear function that computes the annual cost of owning this home if x utility units are used. (ii) What does the y- intercept on the graph of the function represent?
34) An event planner expects about 167 people at a wedding. He knows that this estimate 34) could be off by 30 people (more or fewer). Food for the event costs $57 per person. How much might the event planner's food expense be?
35) The length and width of a rectangle have a sum of 80. What dimensions give the maximum 35) area?
36) A bacteria colony doubles in 5 hr. How long does it take the colony to triple? Use N = N0 36)
2t/T, where N0 is the initial number of bacteria and T is the time in hours it takes the colony to double. (Round to the nearest hundredth, as necessary.)
37) In the formula A = Iekt, A is the amount of radioactive material remaining from an initial 37) amount I at a given time t and k is a negative constant determined by the nature of the material. A certain radioactive isotope has a half- life of approximately 1100 years. How many years would be required for a given amount of this isotope to decay to 35% of that amount?
38) A cone is constructed from a circular piece of paper with a 2- inch radius by cutting out a 38) sector of the circle with arc length x. The two edges of the remaining portion are joined together to form a cone with radius r and height h, as shown in the figure. Express the volume V of the cone as a function of x. plane going from Denver to Los Angeles leaves Denver, which is 850 miles from Los Angeles, at a speed of 510 mph. When they meet, how far are they from Denver?
2
in.
2
in.
39)
An airplane leaves Los Angeles for Denver at a speed of
440
mph. Thirty minutes later, a
39)
40) Two pipes together can fill a large tank in 10 hr. One of the pipes, used alone, takes 15 hr 40) longer than the other to fill the tank. How long would each pipe take to fill the tank alone?
41) A rectangular piece of cardboard measuring 12 inches by 42 inches is to be made into a box 41) with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each such square. What is the maximum volume of this box? If necessary, round to 2 decimal places.
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|
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43) A coil of wire rotating in a magnetic field induces a voltage given by e = 20 sin (Δ4t - ), 43) where t is time in seconds. Find the smallest positive time to produce a voltage of 10 3.
Δ
2
44) The weekly sales in thousands of items of a product has a seasonal sales record 44) approximated by n = 66.47 + 24.6 sin Δt (t is time in weeks with t = 1 referring to the first 24
week in the year). During which week(s) will the sales equal 78,770 items?
45) When a person bends at the waist with a straight back, the force, F, exerted by the person's 45) lower back muscles can be estimated with the formula F = 2.89w cos Ό, where w is the person's weight, and Ό is the angle made by the person's torso relative to the horizontal. Suppose that a 120- pound person bends at an angle of 25°. Determine the force exerted by the person's back muscles, and estimate the angle that requires the back muscles to exert a force of 218 pounds.
1
1
1