sketch a graph where the limit x-->2 exists but f(x) is discontinuous at x=2, what type of discontinuty is this?

State whether the function is a polynomial function or not. If it​ is, give its degree. If it is​ not, tell why not.

4(x-1)^12(x+1)^7

A rectangular box without a lid should be made with 12m2 of cardboard. What are the dimensions of the box that maximize the volume?

a.) 2m x 2m x 2m

b) 1.54m x 1.54m x 0.77m

c) 2m x 2m x 1m

d) 4m x 4m x 2m

A rock is thrown upward from a bridge that is 59 feet above a road. The rock reaches its maximum height above the road 0.9 seconds after it is thrown and contacts the road 2.4 seconds after it was thrown.

Write a function f that determines the rock's height above the road (in feet) in terms of the number of seconds t since the rock was thrown.

f(t)=   

Hint: the function f can be written in the form f(t)=c⋅(t−t1)(t−t2) for fixed numbers c, t1, and t2.

An office supply company manufactures and sells X permanent markers per year at a price of P €/unit. The Price/Demand equation for the markers is: ? = 5 − 0.001?

1.  Write the Revenues function (10%)
2.  What level of production and what price should the company charge for the markers to maximize revenues?

The total cost of manufacturing is: ?(?) = 3000 + 2?

3. Write the Company’s Profit function

4. What level of production and what price should the company charge for the markers to maximize profits?

5. Draw a graph representing the above-mentioned situation

Now the government decides to tax the Company in 1€ for each marker produced. Taking into account this additional cost:

6. Write the company’s new Cost function

7. Write the company’s new Profit function

8. What level of production and what price should the company charge for the markers to maximize profits (with these new conditions)?

   (Please don't skip questions 5, 6, 7, 8, I need especially them)

Problem 1

Part A: Write down an equation of the line L that is passing through the point A(5,4) and is perpendicular to the vector n=<-3,4>.

Part B: Find the unit vector u in the direction of n.

Part C: Find the distance d(Q,L) from the point Q(7, 13) to line L.

Part D: Find the coordinates of the point R on L that is closest to the point Q.

Part E: Now find R by solving the distance minimization problem using single-variable calculus. Which approach do you prefer?

1. What is the name of the method we use to find optimal solutions when given a function and at least one constraint?

2. Give a real-life example of an optimization problem in your academic discipline (a function of several variables with constraints).

3. Write a paragraph about the importance of this topic in science, technology, engineering, and mathematics (STEM) research.

Jaime wants to create a model for the bounce height of a certain brand of Bouncy Ball. After setting the initial velocity at which the ball is thrown and performing several bounces, you and Jaime find that the ball typically reaches a maximum height of 60 inches 0.5 seconds after the first bounce. 1) Assume that the path of the ball is quadratic. Graph the path of the ball from the first bounce until the second bounce. Let t = 0 represent the time at which the ball bounces on the floor for the first time. (t=1 would represent the time that the ball bounces on the floor for second time). PLEASE SHOW WORK

2) Find both of the x-intercepts for the quadratic function that models the relationship between time after the first bounce and height above the floor.

3) What is the vertex of the quadratic function?

4) Use the formula for a quadratic function h(t) = at2 +bt + c along with your 3 data points to solve for a, b and c.

5) Use the zeros of the function to determine the factors of the quadratic function. Write the function in factored form.

6) Write the model of bounce height over time in polynomial form by multiplying out the factors, and including the values found in 4) for a, b and c.

A Tank can hold 200 liters, it currently holds 150 liters of fresh water. 2 different pipes pour in mixture, one at a rate of 1kg of salt per minute at a rate of 6 liters per minute. another at a rate of 2 kg of salt at a rate of 4 liters per minute.

A. If the mixture is flowed out at a rate of 10 liters per minute what is the amount of salt at time t. what about in 10 minutes?

B. the pipes outflow is reduced to 5 liters per minute when will the tank overflow and how much salt will be in it.

C. The tank is over flowing so you decide to turn off the pipe that flows in at 6 liters per minute. how long until the tank returns to 150 liters

plz show work

1. Find Taylor series centered at 1 for f(x) = e^ (x^2). Then determine interval of convergence.

2. Find the coeffiecient on x^4 in the Maclaurin Series representation of the function g(x) = 1/ (1-2x)^2

Find the curvature of the curve Vector r(t)= costi + costj -3sintk at the point (1,1,0)

Q(1) A natural history museum borrows $2,000,000 at simple annual interest to purchase new exhibits. Some of the money is borrowed at 6%, some at 7.5%, and some at 8.5%. Use a system of linear equations to determine how much is borrowed at each rate given that the total annual interest is $137,000 and the amount borrowed at 7.5% is four times the amount borrowed at 8.5%. Solve the system of linear equations using matrices.

Q(2) Use a system of equations to find the cubic function f(x) = ax³ + bx² + cx + d  that satisfies the equations. Solve the system using matrices.

f(−1) = 3

f(1) = 5

f(2) = 30

f(3) = 95

Q(3) An object moving vertically is at the given heights at the specified times. Find the position equation

s = 1/2at² + v₀t + s₀ for the object.

At t = 1 second, s = 136 feet

At t = 2 seconds, s = 108 feet

At t = 3 seconds, s = 48 feet

Solve the given differential equation using an appropriate substitution. The DE is a Bernoulli equation,

A. dy/dx = y(xy^6 - 1)

B. x dy/dx + y = 1/y^2

C. t^2 dy/dt + y^2 = ty

Politics. If 12509 people voted for a politician in his first​ election, 14473 voted for him in his second​ election, and 6752 voted for him in the first and second​ elections, how many people voted for this politician in the first or second​ election?

how do you find the slope of a tangent line?