# Mathematics 106 Problem Set Solutions

1. Find an equation for the line with x - intercept 20 and
y -intercept 5. Sketch its graph.

** Solution :** It goes through (20, 0) and (0, 5), so it has slope
. Its equation

may be written as . Its graph may be drawn

by simply connecting the two intercepts with a straight line.

2. Factor x ^{3} + 3x^{2} − 34x − 120 completely.

** Solution :** Looking at the divisors of 120, we see −4 is a zero of the
polynomial, so

x−(−4) = x+4 is a factor. Factoring x +4 out (using long division , synthetic
division

or some other method ), we get x^{3} + 3x^{2} − 34x − 120 = (x + 4)(x^{2} − x − 30).

The second factor may be further factored at sight or by trial and error to get
x^{3}+3x^{2}−

34x − 120 = (x + 4)(x + 5)(x − 6).

3. For what values of x is negative?

**Solution: **We note the numerator is 0 when x = 0 and when x = −6 and the
denominator

is 0 when x = 2.

When x > 2, all the factors are positive so the quotient is positive.

When 0 < x < 2, x − 2 < 0, so (x − 2)^{3} < 0, but the other factors are positive
so the

quotient is negative.

When −6 < x < 0, x < 0, (x − 2)^{3} is still negative, while x + 6 > 0, so the
quotient is

positive .

When x < −6, x+6 < 0 and x and (x−2)^{3} remain negative, so the quotient is
negative.

We conclude is negative when x < −6 and when
0 < x < 2. We may describe

the set for which is negative as {x|x < −6
or 0 < x < 2} = (−∞,−6) ∪ [ (0, 2).

4. Calculate .

**Solution:** .

5. The distance s (in feet) traversed along a straight
path by an object by time t (in seconds)

is given by the formula s = 8t^{3} + 5t^{2} + 2t.

(a) Find its average speed during the time interval 2 ≤ t ≤ 4.

**Solution:** Its average speed is feet per

second.

(b) Find its instantaneous speed when t = 3.

**Solution:** Its instantaneous speed is . s' = 24t^{2}+10t+2, so its
instantaneous

speed is 248 feet per second.

6. Let f(x) = x^{3} + 5x. Use the definition of a derivative to find f'(x).

**Solution:**

7. Calculate .

**Solution:**

8. Calculate

**Solution:**

9. Calculate .

**Solution: **

10. Write down a strategy for calculating derivatives .

**Solution:** See notes online.

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