Which of the following are subspaces of the vector space of real-valued functions of a real variables? (must select all of the subspaces.)

A. The set of even function (f(-x) = f(x) for all numbers x).

B. The set of odd functions (f(-x) = -f(x) for all real numbers x).

C. The set of functions f such that f(0) = 7

D. The set of functions f such that f(7) = 0

at time t= 0 , a particle is located at the point (4,8,7). it travels in a straight line to the point (7,1,6) has speed 7 at (4,8,7) and constant acceleration 3i-7j-k. find an equation for the position vector r(t) of the particle at time t

1) A force of 2 pounds is required to hold a spring stretched 0.2 feet beyond its natural length. How much work (in foot-pounds) is done in stretching the spring from its natural length to 0.9 feet beyond its natural length?

2) Work of 3 Joules is done in stretching a spring from its natural length to 19 cm beyond its natural length. What is the force (in Newtons) that holds the spring stretched at the same distance (19 cm)?

3) How much work is done lifting a 40-pound object from the ground to the top of a 30-foot building if the cable used weighs 0.5 pounds per foot?
foot-pounds.

Introduce variables to represent items of interest in the problem. For

example, (1) Let n be the number to be found, or (2) Let a be Alice’s

current age, or (3) Let r be the rate of the first car.

(2)

Write down an equation using the information given in the problem. This

is the really important part for this particular assignment!

(3)

Solve the equation and find the value requested in the problem.

The distance between A and B is 180 miles. An automobile at A starts for B are the rate of 40 miles per hour at the same time that an automobile at B starts for A at the rate of 50 miles per

hour. How long will it be before the automobiles meet?

A snail leaves A and travels at the rate of four inches per minute to B. At B the snail catches a ride on a turtle and is carried back the A at the rate of ten inches per minute. If the round trip takes 14 minutes, what is the distance between A and B?

Two boats start at the same point, one goes straight north, the

other straight south. The first boat travels twice as fast as the

second. After three hours they are 72 miles apart. Find the rate

of the faster boat.

In general, can a polynomial function of degree 3 have any absolute extrema? Why?

NOTE- In details with example

A patrol car is parked 50 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of 32 revolutions per minute. How fast is the light beam moving (in ft/sec) along the wall when the beam makes angles of θ = 45°, θ = 60°, and θ = 80° with the line perpendicular from the light to the wall? (Round your answers to two decimal places.)

(a)    θ = 45°

ft/sec

(b)    θ = 60°

  ft/sec

(c)    θ = 80°

ft/sec

Question: What exactly makes the FOIL method similar to using the Distributive Property? Try rewriting an example of binomial multiplication as a combination of two applications of the Distributive Property.

Hint: First, define the Distributive Property, and give examples. Then, tell us what the FOIL acronym means, and give an example. Finally, show how both are equivalent (you could solve the same problem with both methods, and then compare the steps).

"Real-Life" Relationship:You can use the concept behind FOIL to impress your friends.

How? With practice, you can FOIL in your head, without pencil and paper!

First, ask your friends to square a number like 31. Then, set up a FOIL in your mind:

312 = 31*31

=(30 + 1)(30 + 1) [since 31 equals 30 plus 1]

=30*30 + 30*1 + 1*30 + 1*1 [FOIL]

=900 + 30 + 30 + 1

=961

After you get some practice with squaring, you can try other numbers, like 21*19:

21*19

=(20 + 1)(20 - 1) [since 21 equals 20 plus 1, and 19 equals 20 minus 1]

=20*20 - 20*1 + 1*20 - 1*1

=400 - 20 + 20 - 1

=399

Challenge: We can use FOIL to get a formula for (a+b)2. What about (a+b)3, (a+b)4, or (a+b)n, where n is a positive integer?

A store has issued two different coupons for its customers to use. One coupon gives customers $25 off their purchase price, and the other coupon gives customers 35% off of their purchase. The store allows customers to use both coupons and choose which coupon to apply first. For this context, ignore sales tax. Let f be the function that inputs a cost (in dollars) and outputs the cost after applying the "$25 off" coupon, and let g be the function that inputs a cost (in dollars) and outputs the cost after applying the "35% off" coupon.

A.) A customer purchases an item for $120 and asks the cashier to apply the "$25 off" coupon first, followed by the "35% off" coupon. What is the cost of the item after the two coupons are applied?

b.) A customer purchases an item for $120 and asks the cashier to apply the "$25 off" coupon first, followed by the "35% off" coupon. Use function notation to represent the cost of the item (in dollars) after the two coupons are applied.

c.) A customer purchase an item for $120 and asks the cashier to apply the "35% off" coupon first, followed by the "$25 off" coupon. What is the cost of the item after the two coupons are applied?

Consider the function

f(x)=

Express the domain of the function in interval notation:

Find the y-intercept: y=  

.
Find all the x-intercepts (enter your answer as a comma-separated list): x=  

.
On which intervals is the function positive?  

On which intervals is the function negative?  

Does f have any symmetries?

f is even;f is odd;     f is periodic;None of the above.

Find all the asymptotes of f (enter your answers as equations):

Vertical asymptote (left):  

;
Vertical asymptote (right):  

;
Asymptote at

x → ∞

:  

.

Determine the derivative of f.

f'(x)=  

On which intervals is f increasing/decreasing? (Use the union symbol and not a comma to separate different intervals; if the function is nowhere increasing or nowhere decreasing, use DNE as appropriate).

f is increasing on  

.
f is decreasing on  

.

List all the local maxima and minima of f. Enter each maximum or minimum as the coordinates of the point on the graph. For example, if f has a maximum at

x=3 and f(3)=9, enter (3,9)

in the box for maxima. If there are multiple maxima or minima, enter them as a comma-separated list of points, e.g.

(3,9),(0,0),(4,7)

. If there are none, enter DNE.

Local maxima:  

.
Local minima:  

.

Determine the second derivative of f.

f''(x)=  

On which intervals does f have concavity upwards/downwards? (Use the union symbol and not a comma to separate different intervals; if the function does not have concavity upwards or downwards on any interval, use DNE as appropriate).

f is concave upwards on  

.
f is concave downwards on  

.

List all the inflection points of f. Enter each inflection point as the coordinates of the point on the graph. For example, if f has an inflection point at

x=7 and f(7)=−2, enter (7,−2)

in the box. If there are multiple inflection points, enter them as a comma-separated list, e.g.

(7,−2),(0,0),(4,7)

. If there are none, enter DNE.

Does the function have any of the following features? Select all that apply.

Jump discontinuities (i.e. points where the left and right limits exist but are different)Points with a vertical tangent lineRemovable discontinuities (i.e. points where the limit exists, but it is different than the value of the function)Corners (i.e. points where the left and right derivatives are defined but are different)

Upload a sketch of the graph of f. You can use a piece of paper and a scanner or a camera, or you can use a tablet, but the sketch must be drawn by hand. You should clearly indicate all the relevant features of the function, including information that may not have been requested here explicitly, for example the limits at the edges of the domain and the slopes of tangent lines at interesting points (e.g. inflection points).
Make sure that the picture is clear, legible, and correctly oriented. Penalties may apply otherwise.

For the linear system x₁+3x₂=2 3x₁+hx₂=k

Find values for h and k such that the system has:

a) no solution

b) a unique solution

c) infinitely many solutions

Use Cramer's Rule to find the solution of the system of linear equations, if a unique solution exists.

–5x + 2y – 2z = 26

3x + 5y + z = –22

–3x – 5y – 2z = 21

Solve the laplace transform to solve the initial value problem.

y"-6y'+9y=t. Y(0)=0, y'(0)=1

Use Euler's Method with step size 0.11 to approximate y (0.55) for the solution of the initial value problem

   y ′ = x − y, and y (0)= 1.2

What is y (0.55)? (Keep four decimal places.)

How do you know what formula to use when solving an geometry angle question. For example: Jessica sits on top of a 150ft tower. Suddenly she sees a boat from where she sits. If the angle of depression of the boat is 6 degrees, how far is the boat from the base of the tower?

sin 6 degrees= .10
cos 6 degrees =.99
tan 6 degrees =.11

the book used tan 6 degrees to solve, looking for the adjacent side as variable "x".

Why couldn't we use COS? but my other real question is again how would you know which formula to use regarding questions like this and questions not like this.

For an exposure of 43 R (Roentgen), calculate;

a) The charge (in coulombs) liberated per kg of air

b) The number of ion pairs liberated per kg of air

c) The energy absorbed per kg of air

d) The absorbed dose in air