Download Practice Questions for Midterm 1 – Math 1060Q – Fall... The following is a selection of problems to help prepare...

Practice Questions for Midterm 1 – Math 1060Q – Fall 2013
The following is a selection of problems to help prepare you for the first midterm exam. Please note the
following:
• anything taught in this class so far is fair game for the exam.
• there may be mistakes – email [email protected] if you find one.
• learning math is about more than just memorizing the steps to certain problems. You have to understand
the concepts behind the math. In order to test your understanding of the math, you may see questions
on the exam that are unfamiliar, but that rely on the concepts you’ve learned. Therefore, you should
concentrate on learning the underlying theory, not on memorizing steps without knowing why you’re
doing each step. (Some examples of “unfamiliar” problems can be found in the sample problems below, to
give you an idea.)
1. Review your algebra skills! This is crucial, and your grade on the exam will be hurt significantly if you
make algebraic mistakes. Some sample problems are available here:
http://www.math.uconn.edu/~bonik/math1060s13/1060algebra_review.pdf
2. A few more algebraic equations to solve:
6
4
+ 1 = − x−3
(a) − 3x−9
(b)
2
3
7
y
=
2
x+4
(c) 5 −
(d)
(e)
(f)
(g)
y−4
y+2
√
√
√
−4
=
+1=
4
(x+2)(x+4)
y−1
y+1
u+6=u
2w + 4 =
x2
√
5w − 14
+4=4+x
3. A few more algebraic expressions to simplify:
√
4
(a) 48x20 w14
5
1
(b) (c−5 b 3 ) 3
3
1 −5
(c) ( 32
)
3
(d) 81− 4
√
√
(e) (2 2 + 3)(5 10 + 2)
√
√
(f) 6v 8 w 2v 5 w
√
√
(g) 3w 24v 3 + v vw2
(h)
(i)
u2 −4u+4
−5u2 +30u−40
5w2
8ab3
−
a2 u3
4b2 k2
4. Solve the inequality x3 − 4x ≥ 0 and sketch its solution set.
5. Solve the inequality |x − 4| ≤ 10 and write your answer in interval notation.
6. Solve the inequality x2 > 9 and write your answer in interval notation.
7. Solve the equation 4|x + 3| = 12.
8. Solve the equation |5 − x| − 8 = −3.
9. Let f (x) = 5x − 2 and g(x) = x2 + 3x.
(a) What is (f ◦ g)(3)?
√
(b) What is g( x)?
(c) What is g(1 + h)?
(d) What is f −1 (2)?
(e) Is g one-to-one? Why or why not?
(f) Which of the following points is on the graph of f −1 ?
(a) (0,3)
(b) (2,1)
(c) (0,0)
(d) (3,1)
(g) Compute a formula for (g ◦ f )(x) and simplify it.
(h) Compute a formula for f −1 (x).
(i) What is the domain of f /g?
(j) What is the domain of g/f ?
(k) What is the domain of f ◦ g?
(l) What is the domain of g ◦ f ?
√
10. Let f (x) = 2 − x and g(x) = 4/x. Find a formula for each of f + g, f · g, f /g, g/f , f ◦ g, and g ◦ f . In
each case, find the domain of the function.
11. Let f (x) = 1 +
12. Let g(x) =
2
3+x .
x−1
2x+5 .
Find a formula for f −1 .
Find a formula for g −1 .
13. Let f be a one-to-one function such that f (0) = 1, f (1) = 2, f (2) = 3, and f (3) = 4. Let g(x) be a
one-to-one function such that g(0) = 3, g(1) = 2, g(2) = 4, and g(3) = 0.
(a) Find (f ◦ g)(0).
(b) Find (g ◦ f )(0).
(c) Find f −1 (2).
(d) Find g −1 (f (1)).
(e) Find (f −1 ◦ g −1 )(0).
14. Let f (x) = (x − 2)2 + 1. Determine a subset of the domain of f on which f is one-to-one. On this subset
of the domain, find a formula for f −1 .
15. Let g(x) = |x − 3| − 4. Determine a subset of the domain of g on which g is one-to-one. On this subset of
the domain, find a formula for g −1 .
16. Find the equation of a line through (1, 2) and (3, 4).
17. Find the equation of a line passing through (5, 6) that is parallel to the line y + 5 = 3x.
18. What is the equation for the line below?
19. Find the equation of a line that passes through (-3, 2) and is perpendicular to the line y = 5x + 7.
20. For each of the quadratic functions below, do five things:
(a) find its roots, if any
(b) find its y-axis intercept
(c) complete the square
(d) graph it
(e) determine its range
(a) x2 − 4x
(b) x2 + 10x − 4
(c) −x2 + 6x + 2
(d) x2 + 5x + 1
(e) 3x2 − 9x + 12
(f) 2x2 + 3x + 4
21. Find a function f whose graph is a parabola with a vertex at (1, 3) such that f (2) = 7.
22. What is the equation of the parabola below?
23. What is the equation of the parabola below?
24. Sketch a graph of the function f (x) = −|x + 3| − 5, and label three points on the graph, in addition to its
axis intercepts.
√
25. Sketch a graph of the function g(x) = 3x + 9 + 3, and label three points on the graph, in addition to its
axis intercepts.
26. Sketch a graph of the function h(x) = |x2 − 3|, and label three points on the graph, in addition to its axis
intercepts.
27. Compute or simplify the following:
(a) log3 3
(b) log7 1
(c) log25 5
(d) log25
1
5
(e) log25 −5
(f) log25 125
(g) log25 0
(h) log4 237
(i) log2 437
(j) 3log3 9
(k) 3log2 16
(l) eln x
2
(m) eln x
+5
(n) ln e2
(o) ln e0
28. Find an inverse for the function f (x) = 2 log3 x.
29. Find an inverse for the function g(x) =
ex+4
2 .
30. What is the domain of the function ln |x|? Can you find an inverse for this function?
31. Sketch a graph of log2 (x + 4).
32. Solve for x: 3 ln(x) − 2 = 4.
33. Solve for x: log2
1
2
+ 3 = x1 .
x
34. Simplify ln(eln(e ) ).
35. Calculate 3x3 − 2x2 + 4x − 3 ÷ x2 + 3x + 3.
36. Calculate 20x3 + 16x2 + 26x − 9 ÷ 5x − 1.
37. What is the degree and the leading coefficient of the polynomial −3(x − 3)2 (x + 5)(x − 2)4 ? Determine
its end behavior. Sketch a graph of the polynomial.
38. Sketch a graph of the polynomial f (x) = (x − 2)(x2 + 6x + 9).
39. Sketch a graph of the polynomial g(x) = (x − 3)2 (x + 5)(x − 2)3 .
40. Sketch a graph of the polynomial h(x) = −x3 + 2x.
41. Given the function f (x) =
x2 − 9
:
x(x − 3)
(a) Find any vertical asymptotes of f .
(b) Find any horizontal asymptotes of f .
(c) Find any zeros of f .
(d) What is the domain of f ?
(e) Sketch a graph of f .
42. Given the function f (x) =
(x + 1)(x − 4)
:
(x + 4)(x + 1)(x + 7)
(a) Find any vertical asymptotes of f .
(b) Find any horizontal asymptotes of f .
(c) Find any zeros of f .
(d) What is the domain of f ?
(e) Sketch a graph of f .
43. Graph the function f (x) =
1
x−2
+
3
x+2 ,
clearly indicating any zeros, holes, and asymptotes.
44. Express the area of an equilateral triangle as a function of its perimeter.
45. Loudness is measured in decibels. Let’s say that I0 is the level sound that can barely be perceived, and
I0 corresponds to 0 decibels (dB). A cat’s purr has a loudness that is about 316 times I0 , so we could
say the loudness of a cat’s purr is I = 316I0 . Decibels are measured as follows – for a sound of level I,
dB = 10 log10 (I ÷ I0 ).
(a) How many decibels is a cat’s purr?
(b) The record noise level in a sports stadium was set recently by fans of the Seattle Seahawks, at a level
of 136.6 decibels. How many times I0 was the intensity of the noise in that stadium?
46. Kyle is driving from City A to City C at a constant rate of 30 mph. In between, is City B. City A is 180
miles from City C, and City B is 40 miles from City C. Write Kyle’s distance from City B as a function
of time.
47. The length of a rectangle is six feet more than twice its width. Its area is 28. Find the dimensions of the
rectangle.
48. It’s found that the number of apples produced by an apple orchard depends on how densely the trees are
planted – the production goes down if the trees are too close together. When there are 600 trees per acre,
the yield per tree is 8 bushels of apples. For every 50 trees per acre added, the yield is one fewer bushel.
(a) Write a mathematical formula for a function that expresses the total number of bushels per acre
produced when there are n trees per acre.
(b) Graph your function by completing the square and applying graph transformations.
(c) Determine the number of trees per acre the farm should plant in order to maximize their yield.
49. The concentration of a certain antibiotic in a patient’s bloodstream is given by C(t) =
represents the number of hours since the antibiotic was taken.
25t
t2 +300 ,
where t
(a) Explain the meaning of the horizontal asymptote of this function.
(b) What is the concentration after 3 hours on the antibiotic?
(c) The antibiotic is only effective if its concentration is above 0.5. When must the patient take another
dose?
50. Knowing how to calculate is good, and essential in math. But math is also about understanding concepts,
for which you need definitions. So, write a definition of the following terms. You can write it in your own
words or the book’s words, but try to be as specific and accurate as possible.
(a) one-to-one function
(b) inverse function
(c) reciprocal function
(d) composition of functions
(e) graph of a function
(f) domain of a function
(g) range of a function
(h) rational function
(i) polynomial function
(j) linear function
(k) (...getting the idea that this class is all about functions?...)
(l) square root function
(m) absolute value function
(n) quadratic function
51. You could see a few conceptual questions on the exams as well. Here are a few examples.
(a) Explain the difference between f (x) =
x2 −2x+1
x−1
=
(x−1)(x−1)
x−1
and g(x) = x − 1.
(b) True or false? f has a horizontal asymptote at 6 if f (x) → ∞ as x → 6.
(c) Explain why a function needs to be one-to-one in order to have an inverse. If you like, you can use
the function f (x) = |x| as an example.
(d) Explain why the vertical line test tells you if a graph represents the graph of a function or not.
(e) What is the multiplicity of a root of a polynomial? How does knowing the multiplicities of its roots
help you graph a polynomial?
(f) If the range of f is [0, ∞) and the domain of f is (−∞, ∞), what is the domain of f −1 ?
√
(g) Is f (x) = x the inverse of g(x) = x2 ?
(h) How many distinct zeros can a cubic polynomial have?
(i) Complete the following definition: a function f is increasing if, for all x1 < x2 , . . .
(j) Can a polynomial function have a horizontal asymptote? If yes, give an example. If no, explain why.
√
√
(k) Let f (x) = x. Then as x → ∞, x →???.
(l) Define the function log2 x.
(m) Let f (x) = ex , g(x) = 5 + 2x − 4x3 , and h(x) =
f (100), g(100), and h(100).
4x2 −3x+2
x3 −12 .
Place in increasing order the values