Jeopardy Related Rates Extreme Values Optimizat ion Lineari zation $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 Final Jeopardy

$100 Question A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r, of the outer ripple is increasing at a constant rate of 2 ft/sec. When this radius is 4 feet, at what rate is the total area of the disturbed water increasing?

$100 Answer dA/dt = 16π ft^2/sec

$200 Question Two variables x and y are functions of a variable t and are related by the equation x^3 – 2y^2 + 5y = 16 where dx/dy = 4 when x = 2 and y = -1. Find dy/dt at the specific instance.

$200 Answer dy/dt = -16/3

$300 Question A 20 ft long ladder leans against the wall of a vertical building. If the bottom of the ladder slides away from the building horizontally at a rate of 2 ft/sec, how fast is the ladder sliding down the building when the top of the ladder is 12 ft above ground?

$300 Answer dy/dt = -8/3 ft/sec The negative makes sense because the ladder is sliding down the wall.

$400 Question A man 6 ft tall walks at a rate of 5 ft/sec toward a street light that is 16 ft above the ground. At what rate is the length of his shadow changing when he is 10 feet from the light?

$400 Answer dx/dt = -3 ft/sec

$500 Question Water runs into a conical tank at the rate of 9 ft^3/min. The tank stands point down and has a height of 10 ft and base radius of 5 ft. How fast is the water level rising when the water is 6 feet deep?

$500 Answer dh/dt = 1/π ft/sec

$100 Question Find the critical points of f(x) = 2x^3 + 3x^2 – 36x +2.

$100 Answer x = -3, 2

$200 Question Use the first derivative test to find the intervals where f(x) = x^3 – 12x is increasing and decreasing, local and global maximums or minimums. On the interval [0,4].

$200 Answer Increasing (2,4) Decreasing (0,2) Local max (0,0) Global min (2,-16) Global max (4,12)

$300 Question Use the second derivative test to determine the concavity of the function and the relative maximums or minimums.

$300 Answer f’’(2) = 2. + Concave Up x = 2 Minimum

$400 Question Find a point of inflection on the curve f(x) = x^3 – 3x +1.

$400 Answer (0,1)

$500 Question Find all critical points of f(x) = sin^-1 x – 2x. And determine if they are max or mins.

$500 Answer x = √3/2 is a plateau point

$100 Question Find two numbers whose sum is 40 and whose product is as large as possible.

$100 Answer x = 20 and y = 20

$200 Question If a box with a base and open top is to have a volume of 4 cubic feet, find the dimensions that require the least material.

$200 Answer x = 2 ft and y = 1 ft

$300 Question A builder wants to build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen?

$300 Answer x = 50 ft and y = 125 ft

$400 Question An open rectangular box with a square base is to be made from 48 square feet of material. What dimensions will result in a box with the largest possible volume?

$400 Answer x = 4 ft and y = 2 ft

$500 Question A cylinder is inscribed in a cone. As x increases, the volume of the cylinder increases, reaches a maximum value, and then begins to decrease again. Find the dimensions of the cylinder that yield the maximum volume. Also, what is the maximum volume?

$500 Answer r = x = 2 units and h = y = 4 units V= 16π units

$100 Question What is the locally linearization formula?

$100 Answer L(x) = f(a) + f’(a)(x – a)

$200 Question If f’(4) = -3 and f(4) = 6, write an equation for the linearization of x at x = 4. Use this to approximate f(4.1).

$200 Answer y – 6 = -3(x – 4) y ≈ 5.7

$300 Question Explain how you know if the tangent line is an overestimate or underestimate of f(x) at a specific point. Be specific and write your answer in complete sentences.

$300 Answer If the tangent line on a curve is concave down, the tangent line is an overestimate. The tangent line is an underestimate of f(x) when the curve is concave up.

$400 Question Find the linearization at f(x) = cos x sin x at a = π.

$400 Answer y = x - π

$500 Question Find the approximate value of y = √1.02 with linearization of y = √(1 + x)

$500 Answer f(0.02) = … Overestimate C.D.

Final Jeopardy Use MVT to prove a point c = 2 is on f(x) = x^2 on the interval [0,4].