3A.

Find the domain and range of the function. (Enter your answer using interval notation.)

h(x) =

Domain

Range

3B.

Determine whether y is a function of x.

xy + x³ − 2y = 0

Yes, y is a function of x.

No, y is not a function of x.   

It cannot be determined whether y is a function of x.

3C.  

Consider the following function. Find the composite functions

f ∘ g

and

g ∘ f.

Find the domain of each composite function. (Enter your domains using interval notation.)

f(x) =

g(x) = x²

(f ∘ g)(x) =

domain=

g ∘ f)(x)=

domain=

Are the two composite functions equal?

yes or no

3D.

Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator.

−60°

sin(−60°)=

cos(−60°)=

tan(−60°)=

3E.

Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator.

420°

sin(420°)=

cos(420°)=

tan(420°)=

Thank you!

(Optimization problem) You have $500 to buy materials to build a box whose base length is seven times the base width and has no top. If the material for the sides cost $7/m² and the material for the bottom costs $10/m², determine the dimensions of the box that will maximize the enclosed volume. Show all work.

For this problem, consider the function f(x) = x³ - 9x² +15x + 3.

A. Determine the intervals on which f(x) is increasing and intervals on which f(x) is decreasing.

B. Determine all relative (local) extrema of f(x).

C. Determine intervals on which f(x) is concave up and intervals on which f(x) is concave down.

D. Determine all inflection points of f(x).

Determine whether the lines:

L1:x=19+5t,y=7+4t,z=13+3t

and

L2:x=−8+6ty=−17+6tz=−8+6t

intersect, are skew, or are parallel. If they intersect, determine the point of intersection.

Point of intersection ( , , )

I know they intersect, I just don't know where the point is.

Thanks!

Consider the following table of values, where t is the number of minutes and Q ( t ) is the amount of substance remaining in grams after t minutes. t 0 1 2 3 4 5 6 7 8 9 10 11 12 Q ( t ) 300 267 238 212 189 168 150 134 119 106 94 84 75

a. What is the initial value of Q ( t ) ?
b. What is the half-life?

c. Construct an exponential decay function Q ( t ) , where t is measured in minutes.

Q ( t ) =

d. What is Q ( 1 ) ? What does it represent? Round your answer to the nearest integer.
Q ( 1 ) = grams This represents how much of the substance is left after with x minutes.

Given that profit P(x) = −0.3x² +12x −10,

Use the differential of P to approximate the change in profit if production is increased from 15 to 16 units?

Find the actual change in profit when x increases from 15 to 16 units?

Use the first derivative test to find the (x, y) coordinates of the max profit?

1. For the first year, the average smart phone was approximately $250, and sales for that year were forecast to be 997.7 million. For the second year the average smart phone was approximately $200, and sales for that year were forecast to be 1401.3 million. A) Assume that the quantity of smart phones sold each year, q (in millions), is the linear function of price per smart phone, p (in US dollars). Write an equation for q as a function of p. B) Explain, in the context of sales, why it is reasonable that the slope of your linear function is negative. C) In the context of smart phone sales, what is the significance of the verticle axis intercept for your linear function?

In exercises 1–4, verify that the given formula is a solution to the initial value problem.

2. Powers of t.

b) y ′ = t^3 , y(0) = 5: y(t) = (1/5)t^(4) + 5

3. Sines and cosines.

a) x′ = −y, y′ = x, x(0) = 1, y(0) = 0: x(t) = cost, y(t) = sint

1. FIND THE GENERAL SOLUTION TO THE DE: Y”’ + 4Y” – Y’ – 4Y = 0

2. COMPUTE:               L {7 e ^3t – 5 cos ( 2t ) – 4 t ² }                 

3. COMPUTE:              L ^{– 1} {(3s + 6 ) / [ s ( s ² + s – 6 ) ] }           

4. SOLVE THE INITIAL VALUE PROBLEM USING LAPLACE TRANSFORMS:             
5. Y” + 6Y’ + 5Y = 12 e ^t       WHEN :   f ( 0 ) = - 1, AND f ‘ ( 0 ) = 7.                                                                                                                                    

SOLVE THE SYSTEM USING LAPLACE TRANSFORMS:   X’ – X – Y = 1,    AND :  – X + Y’ – Y = 0, WHEN:  X = f ( 0 ) = 0, AND: Y = g ( 0 ) = - ( 5 / 2)   

1.Find an equation for the plane containing the two (parallel) lines v1 = (0, 1, −9) + t(4, 9, −3) and v2 = (9, −1, 0) + t(4, 9, −3).

2.Find a parametrization for the line perpendicular to (6, −1, 1),parallel to the plane 6x + y − 8z = 1,and passing through the point (1, 0, −7).(Use the parameter t. Enter your answers as a comma-separated list of equations.)

Let f(x)=4x^3+21x^2−24x+5. Answer the following questions.

1. Find the interval(s) on which f is increasing.
Answer (in interval notation):

2. Find the interval(s) on which f is decreasing.
Answer (in interval notation):

3. Find the local maxima of f List your answers as points in the form (a,b)
Answer (separate by commas):

4. Find the local minima of f  List your answers as points in the form (a,b)
Answer (separate by commas):

5. Find the interval(s) on which f is concave upward.
Answer (in interval notation):

6. Find the interval(s) on which f is concave downward.
Answer (in interval notation):

Find f.

f ''(x) = 4 − 12x, f(0) = 6, f(2) = 10

A 15 ft ladder leans against a wall. The bottom of the ladder is 3 ft from the wall at time t=0 and slides away from the wall at a rate of 1 ft/sec. Find the velocity of the top of the ladder at time t=3. The velocity of the ladder at time t=3 is --ft/sec