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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(x3), 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 one-to-one, then it has no inverse function.

 


(c) No graph exists for the parametric curve given by:

 


(d) sinh(x) + cosh(x) = ex. (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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