8.

All the real zeros of the given polynomial are integers. Find the zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

P(x) = x³ + 6x² − 32

x =

Write the polynomial in factored form.
P(x) =

9.

Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

P(x) = 4x⁴ − 45x² + 81

x =

Write the polynomial in factored form.
P(x) =

10.

All the real zeros of the given polynomial are integers. Find the zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

P(x) = x³ − 19x − 30

x =

Write the polynomial in factored form.
P(x) =

11.

All the real zeros of the given polynomial are integers. Find the zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

P(x) = x³ − 9x² + 27x − 27

x =

Write the polynomial in factored form.
P(x) =

12.

Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

P(x) = 9x³ + 9x² − x − 1

Write the polynomial in factored form.

13.

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

P(x) = 4x³ + 6x² − 7x − 9

x =

14.

Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

P(x) = 9x³ − 13x + 4

Write the polynomial in factored form.

1- Use calculus to find the absolute maximum and minimum values of the following functions on the given intervals. Give your answers exactly and show supporting work.

f(x) = (7x − 1)e^−2x on [0, 1]

f(x) = x^4 − 2x^2 + 4 on [0, 2]

f(x) = x^3 − 2x^2 + x + 1 on [0, 1]

FInd the limit.

2a) lim (x,y)-->(0,0) (-5x^2)/(2x^2+3y^2)

2b) lim (x,y)-->(0,0) tan(x^2+y^2)arctan(1/(x^2+y^2))

2c) lim (x,y)-->(2,4) (y^2-2xy)/(y-2x)

1. Find the coefficients for expanding y=1 in a Fourier series over the interval [-1,1].
2. Find the coefficients for expanding y=x in a Fourier series over the interval [-1,1].

4. When the ticket price for a concert at the opera house was $50, the average attendance was 4000 people. When the ticket price was raised to $52, the average attendance was 3800 people. a. Assuming the demand function is linear, find the demand function, p. b. Find the number of tickets sold that maximize the revenue. Use the second derivative test to verify it is a maximum. c. Find the price that maximizes the revenue. d. Find the maximum revenue.

5. Find the derivative of ? a. ? = (5? + 3) 2(2? + 1) 4 using the product rule. Factor final answer as much as possible. b. ? = 3? 2−5?+2 4?+1 using the quotient rule. Clean up the numerator but do not factor it.

6. There are two main ways to describe how loud a sound is. One is that you can describe its intensity, I, measured in W/m2, which is the amount of energy per unit time, per unit area, transported by the sound. However, this is not very close to the human experience of sound loudness. The human experience of loudness (actually like the way most of our senses work) is that each factor of 10 in intensity sounds like the same sized “step” in loudness. In other words, in our experience of sound, the difference between 0.01 W/m2 and 0.1 W/m2, seems the same as the difference between 0.1 W/m2 and 1 W/m2. For this reason, when talking about loudness we often use the “decibel scale”, defined by

   ?I? β = (10dB)log I0

   where I is the sound intensity, I0 is a reference intensity and β is the loudness in decibels. A common choice for I0 is the “threshold of hearing”, which for a typical person is I0 = 1 × 10−12 W/m2.

   1. (a) What intensity corresponds to β = 0 dB? Does 0 dB mean “no sound”?

   2. (b) The “threshold of pain” (hopefully the name makes it clear what this means...) is 130 dB. What sound intensity

      does this correspond to?

   3. (c) Some sound has a loudness of 50 dB. Another sound has 1200 times the intensity of the first sound. What is the loudness of the second sound?

1. y = 2x³ - 3x + 1 [-2,2]

2. y = 7 - x² [-1,2]

3. y = e^x [0,ln4]

for each of the following 1-3, is the instantaneous rate of change equal to the average rate of change? If so, where?

4. for each of the following a-c,  find the critical points, determine if there is an absolute max and or min, if so, find them

a. y = 2x³ - 15x² + 24x [0,5]

b. y = x / (x²+3)² on [-2,2]

c. (4x³ / 3 ) + 5x² - 6x on [-4,1]

for parts 1-2 for

use these answers/rules

Find the relative extrema of the function Specifically:

The relative maxima of f occur at x =

The relative minima of f occur at x =

The value of f at its relative minimum is

the value of f at its relative maximum is

Notes: Your answer should be a comma-separated list of values or the word "none".

part 1)

f(x)=9x−(4/x)+6

part 2)

f(x)=(8x^2−7x+32)/(x)

part 3)

Use the derivative to find the vertex of the parabola

y=−3x^2+12x−9

Answer: the vertex has coordinates

x=  

and

y=.

4 parts following these instructions

Find the critical numbers for f and the open intervals on which f is increasing (decreasing)

For the first question, your answer should be a comma-separated list of x values or the word "none". For the other two, your answer should either be a single interval, such as (0,1), a comma-separated list of intervals, such as (-inf, 2), (3,4), or the word "none".

answeres needed for each part

part 1)

Let f(x)=18+3x−x^2

part 2)

let f(x)= 5+2x-x^3

part 3)

let f(x)=6x-6

part 4)

let f(x)= 6x^2-8x^4

PROBLEM: Given f(x)=4x - x² + k & (1,3+k) on the graph of f.
k = 2

a) Write you equation after substituting in the value of k.

b) Calculate the function values, showing your calculations, then graph three secant lines

i. One thru P(1,f(1)) to P₁(0.5,__)

ii. One thru P(1,f(1)) to P₂(1.5,__)

iii. One thru P₁ to P₂                                                                                                                      

c) Find the slope of each of these three secant lines showing all calculations.                        

d) NO DERIVATIVES ALLOWED HERE! Use the results to estimate the slope
of the tangent line at (1,3+k).

e) How would you improve your approximation?                                                  

A trough is filled with a liquid of density 850 kg/m³. The ends of the trough are equilateral triangles with sides 8 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough. (Use 9.8 m/s² for the acceleration due to gravity.)

Find the particular solution to ?″+16?=−24sec(4?) Do not include the complementary solution ??=?1?1+?2?2.

How many X intercept can the graph of a quadratic function have? How many Y intercepts? Explain your reasoning.

Find the first five nonzero terms of the Maclaurin expansion of f(x)=e^-xcos(x)

(Use symbolic notation and fractions where needed.)

Consider the following.

y = −x³ + 6x² + 15x − 9

Give the relative extrema and points of inflection. (If an answer does not exist, enter DNE.)