a) The number of voices I hear at 7am is 12000 at 9am the count is 36000, if the number of voices grows exponentially, find the time it takes to reach 81000 voices?

b) assume one voice tells me to invest 20000 in a south american bank where the interest is compounded continiously, after 6 years i have 24000 dollars, what must the interest rate have been?

Suppose that the mass in a mass-spring-dashpot system with m = 10, the damping constant c = 9, and the spring constant k = 2 is set in motion with x(0) = -1/2 and x'(0) = -1/4.

(a) Find the position function x(t).

(b) Determine whether the mass passes through its equilibrium position. Sketch the graph of x(t).

In the first 5 examples find the derivative of Y

1. Y=epower2x

2. Y=epower-2x +xpower2

3. Y=epowerxpower1/2

4. Y= epower-3/x

5. Y=epower4xdivided by xpower2

Integrate the following

6. Int x epower5xpower2 dx

7. Int epower-2x between 1 and 0

8. Int.epower x1/2 divided by xpower1/2

In general:

1) What are gradient vectors?

a. How are they formed?

b. Where are they located?

c. How are they related to level curves?

d. What direction do they point?

e. Why are they important?

find the curvature and torsion of beta(t) = (e^tcost, e^tsint, e^t)

There is a line through the origin that divides the region bounded by the parabola y=8x−7x^2 and the x-axis into two regions with equal area. What is the slope of that line?

A rectangular box with a square base has a volume of 4 cubic feet. The material for the bottom of the box costs $3 per square foot, the top costs $2 per square foot, and the four sides cost $5 per square foot.

(a) If x is the side length of the square base, and y is the height of the box, find the total cost of the box as a function of one variable.

(b) Find the critical number of the cost function.

(c) Use the Second Derivative Test to show that the critical number for cost is a local minimum

A rectangular box with a square base has a volume of 4 cubic feet. The material for the bottom of the box costs $3 per square foot, the top costs $2 per square foot, and the four sides cost $5 per square foot.

(a) If x is the side length of the square base, and y is the height of the box, find the total cost of the box as a function of one variable.

(b) Find the critical number of the cost function.

(c) Use the Second Derivative Test to show that the critical number for cost is a local minimum

Where is the function increasing or decreasing and what are the relative mins and max?

g(t) = 3t^4-16t^3+24t^2, Domain (-infinity, +infinity)

k(x) = 2x/5 - (x-1)^2/5, Domain [0, infinity)

g(x) = x^3/x^2-3

g(x) = x^2-4√ x

-Use implicit differentiation to find dy/dx where x²y² + x cos(y) = 6

-Use logarithmic differentiation to find the derivative of o(x) = (cos(x))^ln(x)

-Find the first and second derivatives of p(x) = 6 ln(x²+3x)

-A balloon is rising vertically at a constant speed of 5 ft/sec. A boy is

cycling along a straight road at a speed of 15 ft/sec. When he passes

under the balloon, it is 45 ft above him.How fast is the distance between

the boy and the balloon increasing 3 seconds later?

Calculate the directional derivative of f(x,y,z)=x(y^2)+y((1-z)^(1/2)) at the point P(1,−2,0) in the direction of the vector v = 5i+2j−k. (a) Calculate the directional derivative of f at the point P in the direction of v. (b) Find the unit vector that points in the same direction as the max rate of change for f at the point P.

The position of a particle moving along the x-axis is given by x(t) = t^3 + 9t^2 − 21t with t is in [0, 2]. (a) Find the velocity and acceleration of the particle.

(b) For what t-values is the velocity 0? (Enter your answers as a comma-separated list.)

(c) When is the particle moving to the left (velocity is negative)? (Enter your answer using interval notation.)

When is the particle moving to the right (velocity is positive)? (Enter your answer using interval notation.) (

d) What is the farthest the particle gets to the left? What is the farthest the particle gets to the right?

(e) When is the velocity increasing? (Enter your answer using interval notation. If an answer does not exist, enter DNE.) When is the velocity decreasing? (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

(f) What is the maximum velocity of the particle? v =

When a cold drink is taken from a refrigerator, its temperature is 5C. After 25 minutes in a 20C room, its temperature has increased to 10C.

a. what is the temperature of the soda pop after another half hour?

b. how long does it take for the soda pop to cool to 50F?

Fit a quadratic function of the form ?(?)=?0+?1?+?2?2 f ( t ) = c 0 + c 1 t + c 2 t 2 to the data points (0,−1) ( 0 , − 1 ) , (1,8) ( 1 , 8 ) , (2,−7) ( 2 , − 7 ) , (3,−6) ( 3 , − 6 ) , using least squares.