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Math 139A Midterm Exam
1. On your computer, select Internet Explorer, and find the
website:
From here, enter your last name and student ID to log into the Math Placement
test
system . Take test categories 6) Functions, and 7) Trigonometry . You don't need
to
turn in anything for this question: I can access your scores directly.
2. Determine the formula for each of the two functions below:
(a) The function, V (t), representing the value of a home , which was purchased
in
1980 (call this "t = 0") for $80; 000. The value of the home triples every 25
years.
(b) The quadratic function , f(x), containing points (2;18), (0; 0), and (3;3)
3. Let
.
(a) The domain of f(x): Dom(f) =
(b) The domain of g(x): Dom(g) =
(c) The domain of h(x): Dom(h) =
(d) g(f(5)) =:
4. Let f(x) = cos(x^{3}), and g.
For each, state whether the function is:
i) even, ii) odd, or iii) neither.
5. Without a calculator or computer , sketch the approximate graphs of the
functions :
. Indicate which graph below corresponds to which
function. (There's one extra grid if you screw up!)
6. Compute the inverse of the following functions:
(a)
(b)
7. Simplify each of the following to determine it's value:
8. Prove any 2 of the following statements:
(a) Let f(x) be a function with domain . Then the function h(x) given
by: is even, whether f(x) is even, odd, or neither.
(b) If a function f(x) is not onetoone, then it has no inverse function.
(c) No graph exists for the parametric curve given by:
(d) sinh(x) + cosh(x) = e^{x}. (Hint: recall the de nitions of sinh and cosh.)
9. Use Maple for the following problem:
(a) Draw the function f(x) = sin(2x). Include gridlines, and pick a color (your
choice)
for the graph. The view should be: and
. Print the
graph.
(b) On your graph, draw by hand the secant line which intersects your function
at
x = 0 and x = 1. Compute the slope of this secant line.
(c) What is the average rate of change of f (x) over the interval 0≤x≤1?
Explicitly
explain the relationship between this average rate of change and the secant line
your just drew.
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