1. 2x
+ 8
a) Sketch the graph of f(x) = x2 +3 x –
4
2 points – hole and asymptote
1 pt
b) Where is it discontinuous?
2 pts (one point each)
c) At each discontinuity,
does f(x) have a limit, and if so, what is the limit? Justify your answer with
a graph.
4 points (2 points each)
don’t forget notation!
d) On the same graph, sketch
the derivative of the function.
2 points – often graph switched or flipped (asymptotes
should line up)
2. If the position of an object is described as
p(t) = t3 – 2t2
– 8t + 1
a) find the velocity at time
t.
3 points
b) find the acceleration at
time t.
2 points
c) At what times is the
particle at rest?
2 points (use velocity and
set it equal o zero)
d) What is the maximum
velocity?
3 points (derivative = 0
and solve for velocity at that t, or by graphing and using the vetex)
3. Consider f(x) = cos2 x
+ 2 cos x over one complete period beginning with x = 0.
a) Find all values of x in
this period at which f(x) = 0.
4 points (the 2 answers and
proper factoring)
b) Find all values of x in
this period at which the function has a minimum and justify your answer.
4 points (2 points for
derivative, 1 point for setting it to zero, 1 point for checking and solving
for x)
c) Over what intervals in this
period is the curve concave up?
2 points (take derivative
again, set greater than zero and solve)
Could do this with the nDeriv()
fxn on the calculator
4. Let f(x) = 4x3
– 3x – 1.
a) Find the x-intercepts of
the graph of f.
2 points (one for each)
b) Find the derivative of the
graph of f.
3 points (each term)
c) Write an equation for the
tangent line to the graph of f at x = -1.
3 points (1 for finding the
point, 1 for the slope, 1 for plugging it into a linear eqn)
d) Explain why the derivative
is needed for part c.
2 points
5. A man has 340 yards of fencing for enclosing two
separate fields, one of which is to be a rectangle twice as long as it is wide
and the other a square. The square field must contain at least 100 square
yards, and the rectangular one must contain at least 800 square yards.
a) If x is the width of the rectangular field, what are
the maximum and minimum possible values of x?
2 points min
2 points max
1 point units and notation and method
b) What is the greatest number f square yards that can be
enclosed in the two fields?. Justify your answer.
2 points for checking min
2 points for checking max
1 point for checking extrema points
6. Randomness:
a) What
is the domain of √x2 – 9
x – 4
1 point for not dividing by zero
1 point for not taking the even root of a negative
number
1 point for solving for domain correctly
b) For the piecewise
function, f, below, at x = 3 is it
f(x) = x2, x < 3
6x-9, x ≥ 3
a.
undefined
b. continuous
but not differentiable
c.
differentiable but not continuous
d. not
continuous, not differentiable
e.
both continuous and differentiable
3 points
c) Given that if f(x) = g(x)/h(x), then
f’(x) = h(x) g’(x) – g(x) h’(x)
[h(x)]2
and sec x = 1
cos x
Find d tanx.
dx
4 points
6.5 5 points
2 pts power
2 pts trig
2 pt multiplication
application of chain rule
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