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?

Let f(x) = x^3 − x on [−1, 1].

Find the true area of the shaded region using a limit of Riemann sums, taking the sample points to be the left endpoints. Hint: to ease the computations, you can use the fact that (a + b)^ 3 = a ^3 + 3a^2 b + 3ab^2 + b^3 (which follows by expanding the cube).

Verify your work by evaluating the corresponding definite integral using the FTC.

The quantity, q, of a certain skateboard sold depends on the selling price, p, in dollars, so we write q = f(p). You are given that f(100) = 15200 and f '(100) = −90.

(a) What does f(100) = 15200 tell you about the sales of skateboards?

When the price of the skateboard is $_________ , then ______ skateboards will be sold.

What does f '(100) = −90 tell you about the sales of skateboards?

If the price increases from $100 to $101, the number of skateboards sold would _______ (increase/decrease) by roughly________ skateboards

(b) The total revenue, R, earned by the sale of skateboards is given by R = pq. Find R '(p).(chose)

-R '(p) = p 'q

-R '(p) = pf '(p) + f(p)   

- R '(p) = p + q

-R '(p) = q

-R '(p) = f '(p) + f(p)

(c) If the skateboards are currently selling for $100, what happens to revenue if the price is increased to $101?
The revenue  ---Select--- (increases/decreases) by roughly $ _______.

.Suppose I am in a boat and I travel at the bearing N70E at 30 knots for 4 hours. Then, I turn 90 degrees clockwise and travel for 5 hours at the same speed. Find my bearing relative to the dock.

Determine over which intervals the following functions are increasing, decreasing, concave up, and concave down

y = x + sin(πx)

So we have a paraboloid x^2 + y^2 - 2 = z and the plane x + y +z = 1 how do we find the center of mass? For some reason we have to assume the uniform density is 8?

Seems complicated because I don't know where to start?

Water is leaking out of an inverted conical tank at a rate of 11000 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 15 meters and the diameter at the top is 5.5 meters. If the water level is rising at a rate of 25 centimeters per minute when the height of the water is 1.5 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute. Round to the nearest whole number.