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Fall โ06/MAT 150/Worksheet 1 Name: Show all your work.
- (11pts) Use formulas to expand:
a) (3x โ 2)(3x + 2) =
b) (4x โ 5)^2 =
c) (x^2 + 7y)^2 =
d) (x + 4)^3 =
- (8pts) Factor the following. Use either a known formula or a factoring method.
a) x^2 โ 5 x โ 14 =
b) 2x^2 โ 9 x โ 18 =
c) x^3 โ 125 =
- (3pts) Verify the formula for the difference of cubes by multiplying out:
(x โ a)^3 =
- (8pts) Simplify.
a)
x โ 5 x^2 โ 2 x โ 8
2 x x^2 โ 16
b)
x +
2 x โ 6 x โ 1 4 +
x โ 1
- (7pts) Solve the following equations:
a) x^2 = 2x + 24 b) x^2 โ x = 6x โ 20
- (6pts) A 2-lb steak is to be divided among Jeff, Sybill and Christine according to how many buttons there are on their clothing (go figure!). Jeff is wearing 5 buttons, Sybill 6 and Christine 8 buttons, hence Jeff gets 56 of what Sybill gets, and Sybill gets 68 of what Christine gets. How many pounds of steak does each of them receive?
- (6pts) The height of a triangle is 3ft more than the base. What are the base and height if the area of the triangle is 40 square feet?
Fall โ06/MAT 150/Worksheet 3 Name: Show all your work.
- (8pts) Put the following expressions into standard form a + bi:
a) (1 + i)(2i โ 1) + 3i(i โ 1) =
b)
1 + i 5 โ 2 i
c) (justify also) i^42 =
Solve the equations algebraically:
- (5pts) x^4 + 2x^2 โ 35 = 0
- (4pts) | 2 x โ 3 | = 5
Fall โ06/MAT 150/Worksheet 4 Name: Show all your work.
- (5pts) Let f be given by f (x) = (x^2 + x โ 1)x. Find the following values for this function: f (โ1), f (3u), f (x + 4). (Simplify where possible).
- (4pts) Find the domain of f (x) =
5 โ x x โ 2
- (9pts) Use the graph of the function f at right to answer the following questions. a) What is the domain of f? b) Find f (4) and f (โ1). c) List the x-intercepts of the graph. d) Where is the function decreasing? e) What are the solutions of the equation f (x) = โ4? f) Where is f (x) > 0?
- (8pts) The function f (x) = x^4 โ 4 x^2 โ 3 x + 4 is given. a) Sketch the graph of f on paper. b) List the numbers where f has a local minimum or maximum. What are the local minima (i.e. the y-values)? Accuracy: 3 decimal points. c) List the intervals where f is increasing. d) What is the range of this function?
- (4pts) The function f (x) = x^5 โ 4 x^3 + 7x is given. a) Determine algebraically whether this function is even, odd or neither. b) Graph the function on paper. Does the graph support your conclusion from a) and why?
- (6pts) The function f is given below. a) Sketch the graph of f on paper. b) Find the domain and range of f.
f (x) =
โ 2 x โ 3 , if โ 4 โค x โค 3 x โ 1 , if 3 < x < 7.
- (5pts) Use the basic graph of y = (^1) x and transformations to help you sketch the graph of y = (^) x+3^2. Explain how you transform the original graph and what the asymptotes of the new graph are.
- (7pts) The graph of the function f is given below. On three separate graphs, sketch the graphs of the functions f (x) + 2, f (2x) and โf (x โ 2). Label all the relevant points.
Fall โ06/MAT 150/Worksheet 6 Name: Show all your work.
- (4pts) The table below indicates the values of f (x) and g(x) for certain numbers. Find the requested composites at right.
x -2 1 4 7 9 f(x) 9 -2 1 7 4 g(x) 4 9 7 1 -
(f โฆ g)(4) =
(g โฆ f )(7) =
(f โฆ f )(โ2) =
(g โฆ g)(1) =
- (8pts) Let f (x) =
x + 7 and g(x) = x^2 + x โ 1. Find the following composites and simplify where possible:
(f โฆ g)(x) =
(g โฆ f )(x) =
(g โฆ g)(x) =
- (4pts) Find functions f and g so that f โฆ g = H, if H(x) =
x + 3
. Find two different
solutions to this problem, neither of which is the โstupidโ one.
Fall โ06/MAT 150/Worksheet 7 Name: Show all your work.
- (4pts) Evaluate without using the calculator:
log 5 125 = log 8 641 = log 4
2 = loga^4
a^3 =
- (6pts) Solve the equations:
log 2 (2x + 5) = 4 103 xโ^1 = 32
- (3pts) Write as a sum and/or difference of logarithms. Express powers as factors. Simplify if possible.
log 3
92 xโ^3 โ x + 7
- (3pts) Write the following as a single logarithm. Simplify if possible.
3 2 log^ x
(^12) + 2 log x (^11) =
- (2pts) Compute the following number using your calculator. Show how you obtained your number.
log 7 14 =
- (5pts) Solve the equation:
log 2 (x โ 3) + log 2 (x โ 1) = 3
- (7pts) At an archaelogical dig, the remains of a person were found. Test indicated that the amount of carbon 14 in their body was 30% of the original amount. How long ago did this person die? (The half-life of carbon 14 is 5600 years.)
- (5pts) Draw two periods of the graph of y = 3 cos(4x). What is the amplitude? The period? Indicate where the special points are (x-intercepts, peaks, valleys).
- (5pts) A ship, offshore from a statue known to be 70ft tall, takes a sighting of the top of the statue. If the angle of elevation is 12โฆ, how far offshore is the ship?
- (3pts) Use the unit circle to find the domain of cot ฮธ.
Fall โ06/MAT 150/Worksheet 9 Name: Show all your work.
- (6pts) Without using the calculator, find the exact values of the following inverse trigonometric functions. Draw the unit circle and the appropriate angle under the expression.
arcsin
= arccos
= arctan
- (3pts) Use a picture to find the exact value below. Do not use the calculator.
arccos
cos
6 ฯ 5
- (4pts) Find the exact value of the expression below. Draw a picture and do not use the calculator.
sin(arctan(โ4)) =
