The temperature in Gavin's oven is a sinusoidal function of time. Gavin sets his oven so that it has a maximum temperature of 320°F and a minimum temperature of 260°. Once the temperature hits 320°, it takes 20 minutes before it is 320° again. Gavin's cake needs to be in the oven for 30 minutes at temperatures at or above 310°. He puts the cake into the oven when it is at 290° and rising. How long will Gavin need to leave the cake in the oven? (Round your answer to the nearest minute.)

____ min

For each of the following mathematical functions (or equations):

(i) Take the first derivative dy/dx,

(ii) Set dy/dx = 0, then solve for x .

(iii) Take the second derivative d(dy/dx)/dx. Is the second derivative positive or negative at x*? Is this a relative minimum point or a relative maximum point? Or neither?

1. 1) Y= 1500 X – (41,000,000 + 500 X + .0005 X2)

2. 2) Y= 12,100,000 + 800X + .004 X2 X

3. 3) Y=(1800-.006X)X

4. 4) Y=1800X-.006X -(12,100,000+800X+.004X )

5. 5) Y= (4) (5000) + 50 (5000/X) + (.5) (X/2)

6. 6) y=x3 –12x2 +36X+8

in the triangle PQR, the point S divides the line PQ in the ratio 1:3, and T divides the line RQ in the ratio 3:2.

PR =a and PQ =b

express the following in terms of a and b

PS , SR, and TQ

Find a basis for the subspace of Pn defined by V={p an element of Pn, such that p(1)=0}. What is the dimension of V?

1. Find the equation of the line with slope of m = −" # and through the point (8,−1).Write your final answer in slope -intercept form. ______________________________

2. Find the equation of the line through the points (2,−7) ??? (−4,−8 ). Write your final answer in slope intercept form. ______________________________

3. Find the equation of the line which passes through the point (10,−3) & is perpendicular to the line 5?+3?=2 ______________________________

Show that ? = {1 − ?, 2 + ?2, 1 + ? − ?2} forms a basis for ?₂(ℝ)

Find the equation (in terms of x and y) of the tangent line to the curve r=5sin5θ at θ=π/3

Find the local maximum and minimum values and saddle point(s) of the function.

f(x,y)=5-10x+12y-5x^2-4y^3

Find the Taylor series or polynomial generated by the following functions

a. )f(x) √ x centred at x=4 , of order 3

b.) f(x) cosh x= e^x+e^-x/(2), centred at x=0

c.) f(x) = x tan^-1x^2 , centred at x=0

d.) f(x) = 1/(√1+x^3) , centred at x=0 , of order 4

e.) f(x) = cos(2x+pie/2) centred at x= pie/4

We consider a rectangular parallelepiped-shaped box based on a rectangle and open from above. The height of the box is 4 dm. The base of the box has a fixed perimeter 20 dm and one side of it is x with 0 <x <10.
a. Prove that the total area of the box as a function of x is E(x)=-x²+10x=80, x belongs to (0,10)
b. Find for which value of x the box has a maximum area.
c. Show that E^'[E(X) / 40] <6, for each x belongs to (0,10).
d. Find the value of x for which it holds E(x-2)=absolute value of x-7 + 105.

Find the Taylor series for f (x) = 2/(1+3x) at x = a where a = 0

9) R is the region bounded by the curves ? = x^3 , y=2x+4 , And the y-axis.

a) Find the area of the region.

b) Set up the integral you would use to find the volume of a solid that has R as the base and square cross sections perpendicular to the x-axis.

The National Bank, like most other large banks, found that using automatic teller machines (ATMs) reduces the cost of routine bank transactions. National installed an ATM in the corporate offices of the Fun Toy Company. The ATM is for the exclusive use of Fun's 595 employees. After several months of operation, a sample of 100 employees revealed the following use of the ATM machine by Fun employees in a month:

a. What is the estimate of the proportion of employees who do not use the ATM in a month?

Proportion %:

b-1. For estimate of the proportion of employees who do not use the ATM in a month, develop a 80% confidence interval. (Round the final answers to 3 decimal places.)

80% confidence interval is _____and ______ .

b-2. Can National be sure that at least 40% of the employees of Fun Toy Company will use the ATM?

Yes or No?

c. How many transactions does the average Fun employee make per month? (Round the final answer to 2 decimal places.)

Total transactions ___________

d. Develop a 80% confidence interval for the mean number of transactions per month. (Round the final answers to 3 decimal places.)

80% confidence interval for the mean number of transaction per month is ____ and ______ .

e. Is it possible that the population mean is 0?

Yes or No?

A cereal box, in the shape of a rectangular prism and with a closed top, is to be
constructed so that the base is twice as long as it is wide. Its volume is to be 8000cm3。

Find the dimensions that will minimize the amount of cardboard required to make the box.