Math 34A Practice Final

Math 34A Practice Final Problems

1)Find values of x where the function f(x) = x4 – 4x3 + 4

a) is increasing

b) is concave down

c) has a minimum point

2) A theater currently sells tickets at a price of $8, and sells 1000 tickets. It is estimated that for each 10-cent increase in the price, 20 fewer tickets will be sold.What is the optimal price for the theater to charge (to maximize their income)?

3) Given the two functions f(x) = 3x2 – 8x and g(x) = -2x2 + 5

a) Find the equation of the line tangent to f(x) at x=2

b) For which value of x is the slope of f(x) equal to the slope of g(x)?

4) A rock is dropped from the top floor of a tall building. Its height above the ground (in meters) is given by the formula h(t) = 500-5t2, where t is the time in seconds after the rock is thrown.

a) When does the rock hit the ground?

b) What is the speed of the rock at the time found in part a)?

c) How tall is the building?

5) The table gives the position of a car at various times during an afternoon drive.

a) Find the average speed of the car between 2pm and 4pm.

b) During what time interval was the average speed the greatest?

6) A cylindrical container with no top is to be made to hold a volume of 12π cm3. The material for the sides costs 2 cents per cm2 and the material for the bottom costs 3 cents per cm2. What are the dimensions of the least expensive container?

Radius = ______cm

Height = ______cm

7) Given m(t) = 3e5t – k2π + t-2

a) Find m’(t)

b) Find m”(2)

8) An initial deposit of $2000 is made into a bank account that yields 6% interest, compounded annually. Find an expression for the number of years until the account balance will be $10,000. You may leave logarithms in your answer.

9) A certain radioactive substance decays exponentially. If 500 grams of the substance were present initially and 400 grams are present 50 years later, how many grams will be present after 200 years?

10) A farmer wants to build a fence around a rectangular field. One side of the field will border a river, so that side will require no fencing. The fence costs $10 per meter, and he has a total of $4000 to spend. Find the dimensions of the field that will enclose the largest area.

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Math 34A Practice Final Problems– Answers

1a) f(x) increasing for x>3

1b) f(x) concave down for 0<x<2

1c) f(x) has minimum when x=3

2) $6.50

3a) y=4x-12

3b) x=4/5

4a) 10 sec

4b) 100 m/s

4c) 500 meters

5a) 30 mph

5b) 1:30pm to 2pm (avg speed is 50mph)

6) Radius = 2cm

Height = 3cm

7a) m’(t) = 15e5t – 2t-3

7b) m’’(2) = 75e10 + 3/8

8) t = ln(5)/ln(1.06)

t = log(5)/log(1.06) is also correct

9) 204.8 grams

10) 100m wide (perpendicular to river), 200m long (along the river)