100
Math 170 – 01: BQ#1 (thru 1.6)
Name:
Show as much work as possible to receive full credit. Give clear, concise explanation when asked to explain anything.
1. Suppose P ( x ) is a polynomial function of degree 5.
(2 points each)
(a) What is the domain of P?
(b) How many x-intercepts (zeros, roots) can P have? Give all possible answers.
(c) How many y-intercepts can P have?
(d) Is it possible that P ( x ) £10 for all x? Explain.
2. Suppose P ( x ) is a polynomial function of degree 4.
(2 points each)
(a) What is the domain of P?
(b) How many x-intercepts (zeros, roots) can P have? Give all possible answers.
(c) How many y-intercepts can P have?
(d) Is it possible that P ( x ) £10 for all x? Explain.
3. Let f ( x ) =
(8 points)
p ( x)
2
3
. Show that f is a rational function by writing it as f ( x ) =
.
+
q ( x)
x x +1
4. Find the domain of each of the following functions.
(a) f ( x ) =
x + 5 - ln (8 - x )
x2 - 9
(b) f ( x ) = tan ( x )
(c) f ( x ) = csc ( 4x - 9)
(8 points each)
4. Suppose P ( x ) = the profit (in $trillion) earned from the sale of x million SUVs. Explain
what each of the following statements means in context. (4 points each)
(a) P (15) = 2.17
(b) P¢ (15) = 0.306
5. The graph below shows the percentage of U.S. residents that had access to Broadband (highspeed) internet x years after 2000. Use a tangent line to estimate the derivative at x = 5.
Explain what this represents in the context of this function. (8 points)
6. Suppose that f is a function such that f (3) = 5 and -2 £ f ¢ ( x ) £ 4 for all x. Give all possible
values for f ( 7) .
(8 points)
7. Suppose g ( x ) = f ( x ) + 6 .
(a) How are the graphs of f and g related to each other?
(4 points)
(b) How are the graphs of f ¢ and g¢ related to each other?
(4 points)
8. Below is shown the graph of y = f ¢ ( x ) . Use it to answer the following.
(4 points each)
(a) Where in the interval [-2, 4] is f decreasing?
(b) Where in the interval [-2, 4] is f concave down?
(c) Where does f have stationary points?
(d) Where does f have local minimums?
(e) Suppose f (1) = -2 . Give the equation for the tangent line to the graph of f at x = 1.