Let the function c(v) model the gas consumption (in liters/km) of a car going at velocity v (in kilometers/hour). In other words, c(v) tells you how many liters of gas the car uses to go 1 km, if it is going at velocity v.

You find that (80) 0.04 and '(80) 0.0004

1. Let the function d(v) model the distance the same car goes on 1 L of gas at velocity v.

a. Express the relationship between c(v) and d(v) in an equation. [4 pts]

b. Find d(80) and d’(80). (Hint: Find the general d’(v) first.) [4 pts]

c. Interpret your result for d’(80) in a sentence. (That is, “When the car is travelling at 80 kph ….” ) [4 pts] (Even if you couldn’t get part b, you can still tell me what d’(80) means about the car.) [5 pts]

Calculate both of the Taylor series of:

- tanx up to and including the x^7 term

- secx up to and including the x^6 term.

**Please answer in full. If you cannot complete because the "derivatives are getting too complicated" then please do not attempt this question. I need a FULL answer. I keep getting stuck, hence the reason why I need a FULL answer.

A water trough is 8 m long and has a cross-section in the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 70 cm wide at the top, and has height 40 cm. If the trough is being filled with water at the rate of 0.3 m³/min how fast is the water level rising when the water is 20 cm deep?

Find the equation of the tangent line at x=2 to the graph of

y= x^2-x-7

Write your answer as a simplified slope-intercept equation y=mx+b.

For example   y=7x-8

Use a definition of power series, to find he first 5 terms of a Taylor series of f(x) at a =9 for

f(x)=root of x

For exercise 10, find all solutions exactly on the interval 0 ≤ θ < 2π.

10. cot x + 1 = 0

For the following exercises, solve exactly on [0, 2π)

13. 2cos θ = √2
16. 2sin θ = − √3

19. 2cos(3θ) = −√2

22. 2cos (π/5 θ)= √3

The Koch snowflake is formed by making an equilateral triangle of 3 congruent Koch curves at each stage of the iteration. The perimeter of this snowflake is infinite based on the Koch curve results, while its area is finite.

Convince the reader that the following is true. Justify with algebra that (1) the Koch snowflake has an infinite perimeter as n approaches infinity and that (2) the Koch snowflake has a finite area as n approaches infinity.

A tugboat goes 24 miles upstream and 28 miles downstream in a total of 8 hours on a river that has a current of 3 mph. Find the speed of the tugboat in still water.

a) A full conical water tank has height 3 m and diameter across the top 2 m. It is leaking water at a rate of 10,000 cm^3 per minute. How fast is the height of the water decreasing when the volume of water is 3,000,000 cm^3 ?

b) Two planes are approaching the same air traffic control tower at equal and constant altitude. Plane A is heading due north at 200 mph, while Plane B is heading due west at 175 mph. At what rate is the distance between the two planes closing when plane A is 60 mi away and plane B is 120 mi away?

c) A spherical balloon has radius 75 cm. Use differentials to estimate the volume of rubber given that the rubber is 1 mm thick.

d) Find the absolute maximum and minimum values of f(x) = 2x^3 − 3x^2 − 12x + 1 on the closed interval [−2, 3].

e) Evaluate lim t→0 (e^2t − 1) / sin(t)

Thank you!

) The air in a room with volume 210m3 contains 0.2% carbon dioxide by volume initially. Fresher air with only 0.1% carbon dioxide flows into the room at the rate of 3m3/min and the mixed air flows out at the same rate. Find the percentage of carbon dioxide in the room as a function of time. What is the percentage of carbon dioxide in the room at the end of 30 minutes? What happens in the long run(that is what happens to your function when time t goes to infinity)?

1). Find the dervatives dy/dx and d^2/dx^2 , and evaluate them at t = 2.

x = t^2 , y= t ln t

2) Find the arc length of the curve on the given interval.

x = ln t , y = t + 1 , 1 < or equal to t < or equal to 2

3) Find the area of region bounded by the polar curve on the given interval.

r = tan theta , pi/6 < or equal to theta < or equal to pi/3

4) Find the length of the polar curve on the given interval

r = (theta)^2 , 0 < or equal to theta < or equal to pi/2

Last few questions from my math packet, any help would help ! Thank you so much.

An object is taken out of a 65 Froom and placed outside where the temperature is 35 F

Five minutes later, the temperature is 63 F It cools according to Newton's Law.

a) Set up the IVP for this problem.

b) Solve the IVP from (a).  

c) Use (b) to determine the temperature, to the nearest tenth, of the object after one hour.

2.       A tank initially contains 120 L of pure water. A salt mixture containing a concentration of 1.5 g/L enters the tank at a rate of 2 L/min, and the well-stirred mixture leaves the tank at the same rate. Find an expression for the amount of salt in the tank at any time t. Also, find the limiting amount of salt in the tank as t →∞.

Find the volume of the solid obtained by rotating the region bounded by y = x 3 , y = 1, x = 2 about the line y = −3.

Sketch the region, the solid, and a typical disk or washer (cross section in xy-plane).

Show all the work and explain thoroughly.