Show your work to receive credit! Use proper mathematical
sentences (= signs, parentheses,
etc.), and be sure to spell out a clear answer to every question asked.
If
you depend on the calculator for an answer, explain exactly what buttons you
pushed.
1. Find all real solutions of each equation:
2. The solid line contains the point (4, 1) and is
perpendicular to the dotted line whose
equation is y = 2x.Give the equation of the solid line in slope - intercept form .
3. Give the solution set of each inequality in interval
notation.
4. Graph the quadratic function f(x) = 12x − x2 using its
vertex, axis of symmetry
and intercepts. Label the vertex and intercepts by their coordinates (x, y) in
the plane.
5. Consider the function f whose graph is as follows:
a) De termine the y -intercept, and list all x-intercepts.
b) Determine all intervals where f(x) is strictly positive
(f(x) > 0).
c) Determine whether f is increasing, decreasing, or
neither, on the interval (1, 3).
d) Find all numbers x at which f has a local minimum. What
is, in each case, the
value y of the local minimum?
6. Plot the function 
by starting with the graph of a basic function
and then using the techniques of shifting, stretching and/or reflecting. (Make a
separate
graph for each intermediate step , labelled by a formula for the intermediate
function.)
7. Consider the polynomial f (x) = (2x + 3)(x + 1)2(x −
1)2(2x − 3).
a) Like what monomial does f(x) behave for large values of |x|?
b) List all x-intercepts of f in increasing order .
c) Sign chart : find in which intervals of the x-axis f(x) is positive (resp.
negative).
d) Sketch the graph of f. Can you identify any local
maxima?
8. For each of the fol lowing functions , compute the
difference of degree between numerator
and denominator . Deduce whether the end behavior is a horizontal asymptote,
an oblique asymptote, or like a monomial. For extra credit, find the equations
of any
asymptotes.
degree(numerator) − degree(denominator) =
End behavior:
degree(numerator) − degree(denominator) =
End behavior:
degree(numerator) − degree(denominator) =
End behavior:
9. Consider the function
(obtained by long division).
a) Find the domain of f, and the equations of all asymptotes of the graph.
b) Find all x- and y-intercepts.
c) Find any point (x, y) where the graph touches or crosses an asymptote.
d) Sign chart: find in which intervals of the x-axis f(x) is positive (resp.
negative).
e) Sketch the graph of f. Display and label all elements
found under a)–c).
10. a) Find any rational zeros of f(x) = x3 − 6x − 4.
b) Now factor f (x) (using long division if necessary ) and find all its zeros.
11. Solve the following equations. (Give exact solutions, using logarithms if
necessary.)
12. You borrow one cent at the usurious interest rate of
20% per year. How much do
you owe back after 50 years, to the nearest cent, if the interest is compounded
a) Yearly?
b) Daily?
c) Continuously?
