A curve c is defined by the parametric equations

x= t^2 y= t^3-4t

a) The curve C has 2 tangent lines at the point (6,0). Find their equations.

b) Find the points on C where the tangent line is vertical and where it is horizontal.

13. Minimizing Perimeter: What is the smallest perimeter possible for a rectangle whose area is 36 in2, and what are its dimensions? Also, provide the perimeter and area equations. SHOW WORK.

Perimeter Equation: _________________Area Equation: ___________________

Dimensions (include units): ___________Perimeter (include units):___________

14. Find the linearization L(x) at x = 2 of the function, f(x)=√(x^2+12). SHOW WORK.

Linearization L(x): ______________________

You must use limit theory for your answers to this question, results obtained by the algorithms of calculus will not earn credit.

1) The height, ?, in metres, (with ? measured in seconds) of a cricket ball (on being mishit straight upwards) is modelled by the equation:

?(?) = 25? − 5?²

a) Show how to derive an expression for the instantaneous velocity at ? = ? seconds.

Simplify your expression as far as possible.

b) What is the initial velocity?

c) Find the instantaneous velocity when ? = 3

d) What does your answer to part (c) mean?

e) What will be the instantaneous velocity when the cricket ball reaches its maximum

height?

f) What is the maximum height reached by the cricket ball?

A gourmet bottled tea drink manufacturer has found that the cost to produce 150 cases of bottled tea is $10,000 while it costs $13,150 to produce 325 cases of bottled tea.

a. Find the linear cost function, C(x) where x is the number of cases of bottled tea.

b. If the revenue function for selling x cases of teach is R(x) = 40x, how much does each case of bottled tea sell for?

c. How many cases of bottled tea do they need to sell each month to break even? Round appropriately. Include correct units.

d. If the profit was $1714, how many cases were sold?

*Please explain how you got the answer and show the work, that would be very helpful!*

Evaluate the line integral, where C is the given curve.

xe^yzds, Cis the line segment from

(0, 0, 0) to (2, 3, 4)

(a) Suppose that f is a polynomial of degree 3 or more. Explain, in your own words, how you would

use real zeros of f to determine the open intervals over which f(x) > 0 or f(x) < 0. Be brief

and precise. In particular, you need to tell how and where the sign of f changes.

(d) Rewrite the expression (cos x)2x in terms of natural base e.

4. Let f(x) = x2 + 5. Find lim_h-0 f(3+h) - f(3)/h

Find the volume of the solid obtained by rotating the region bound by the line x=0 and the curve x=4-y² about x=-1

Imagine you are writing questions for an algebra course. You want to write questions that include finding the zeros of polynomials. To balance the difficulty level of the test’s questions, you decide to include two different types of questions, as described here:

• Question #1: A polynomial of degree at least 3 where all the zeros are positive whole numbers
• Question #2: A polynomial of degree at least 3 where one or more of the roots are fractions

Find two polynomials for the two described questions. Explain how you know each question satisfies the requirements stated. What approach did you use to find these polynomials?

How would you find the absolute maximum and the absolute minimum over the interval of :

f(x)=x²+2.5x-6, -5 ≤ x ≤ 5

f(x)=12(1.5^x)+12(0.5^x), -3 ≤ x ≤ 5.1

1. All of the following regions are given in rectangular coordinates. Give a sketch of the region and convert to a coordinate system, which you believe would be the most convenient for integrating over the given region, and a brief explanation as to why you chose that coordinate system.

d) E = {(x, y, z) | − 2 ≤ x ≤ 2, − √ 4 − x^2 ≤ y ≤ √ 4 − x^2 , 2 − sqrt4 − x^2 − y^2 ≤ z ≤ 2 + sqrt4 − x^2 − y^2}

Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)

cos²(θ) − cos(θ) − 12 = 0

The number of cell phone subscribers in the United States between the years 2000 and 2010 is approximated by the function N(t) = 385.474 1 + 2.521e−0.214t (0 ≤ t ≤ 10) where N(t) is measured in millions and t is measured in years, with t = 0 corresponding to the year 2000.† How many cell phone subscribers were there in the United States in 2000? (Round your answer to one decimal place.) million subscribers.

If the trend continued, how many subscribers were there in 2011? (Round your answer to one decimal place.) million subscribers

Use the power series method to solve the given initial-value problem. (Format your final answer as an elementary function.)

(x − 1)y'' − xy' + y = 0, y(0) = −4, y'(0) = 7

y =

f(x)=x4−8x2

a. Interval(s) of increase/decrease
b. Local maximum and minimum values as coordinates (x,y)
c. Intervals of concavity
d. Inflection points as coordinates (x,y)
e. Y-intercepts as coordinates
f. X-intercepts as coordinates

Is the interval [-4, 7) open, closed, half–open, or unbounded?