(a) Use a graph of P(q) = 6qq2 to determine whether each of the following derivatives is

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 3x

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 5

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = x12

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = x12

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 8t3

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = x4/3

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 5x + 13

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 3t4 2t2

For Problems 138, find the derivative. Assume a, b, c, k are constants. f(q) = q3 + 10

For Problems 138, find the derivative. Assume a, b, c, k are constants. f(x) = 1 x4

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 6x3 + 4x2 2x

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = x2 + 5x + 9

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 8t34t2+12t3

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 3x2 + 7x 9

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 3x4 4x3 6x

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 4.2q2 0.5q + 11.27

For Problems 138, find the derivative. Assume a, b, c, k are constants. f(z) = 1 z6.1

For Problems 138, find the derivative. Assume a, b, c, k are constants. g(t) = 1 t5

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = x

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 1 r7/2

For Problems 138, find the derivative. Assume a, b, c, k are constants. f(x) = 3 1 x3

For Problems 138, find the derivative. Assume a, b, c, k are constants. h() = 1 3

For Problems 138, find the derivative. Assume a, b, c, k are constants. z = (t 1)(t + 1)

For Problems 138, find the derivative. Assume a, b, c, k are constants. R = (s2 +1)2

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = z2 + 1 2z

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 3t5 5t + 7 t

For Problems 138, find the derivative. Assume a, b, c, k are constants. h(t) = 3 t + 4 t2

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = 3t2 + 12 t 1 t2

For Problems 138, find the derivative. Assume a, b, c, k are constants. h() = (1/22)

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = x(x + 1)

For Problems 138, find the derivative. Assume a, b, c, k are constants. y = ax2 + bx + c

For Problems 138, find the derivative. Assume a, b, c, k are constants. f(x) = kx2

For Problems 138, find the derivative. Assume a, b, c, k are constants. v = at2 + b t2

For Problems 138, find the derivative. Assume a, b, c, k are constants. Q = aP2 + bP 3

For Problems 138, find the derivative. Assume a, b, c, k are constants. V = 4 3r2b

For Problems 138, find the derivative. Assume a, b, c, k are constants. P = a + bt

For Problems 138, find the derivative. Assume a, b, c, k are constants. h(x) = ax + b c

For Problems 138, find the derivative. Assume a, b, c, k are constants. w = 3ab2q

(a) Use a graph of P(q) = 6qq2 to determine whether each of the following derivatives is positive, negative, or zero: P(1), P(3), P(4). Explain. (b) Find P(q) and the three derivatives in part (a).

Let f(x) = x3 4x2 + 7x 11. Find f(0), f(2), f(1).

Let f(t) = t2 4t + 5. (a) Find f(t). (b) Find f(1) and f(2). (c) Use a graph of f(t) to check that your answers to part (b) are reasonable. Explain.

Find the rate of change of a population of size P(t) = t3 +4t +1 at time t = 2.

The height of a sand dune (in centimeters) is represented by f(t) = 700 3t2, where t is measured in years since 2005. Find f(5) and f(5). Using units, explain what each means in terms of the sand dune.

In Problems 4445, find the relative rate of change f(t)/f(t) at the given value of t. Assume t is in years and give your answer as a percent. f(t) = 3t + 2; t = 5

In Problems 4445, find the relative rate of change f(t)/f(t) at the given value of t. Assume t is in years and give your answer as a percent. f(t) = 2t3 + 10; t = 4

The number, N, of acres of harvested land in a region is given by N = f(t) = 120t, where t is the number of years since farming began in the region. Find f(9), f(9), and the relative rate of change f/f at t = 9. Interpret your answers in terms of harvested land.

Zebra mussels are freshwater shellfish that first appeared in the St. Lawrence River in the early 1980s and have spread throughout the Great Lakes. Suppose that t months after they appeared in a small bay, the number of zebra mussels is given by Z(t) = 300t2. How many zebra mussels are in the bay after four months? At what rate is the population growing at that time? Give units.

The quantity, Q, in tons, ofmaterial at amunicipal waste site is a function of the number of years since 2000, with Q = f(t) = 3t2 + 100. Find f(10), f(10), and the relative rate of change f/f at t = 10. Interpret your answers in terms of waste.

If f(t) = 2t3 4t2 + 3t 1, find f(t) and f(t).

If f(t) = t4 3t2 + 5t, find f(t) and f(t).

Find the equation of the line tangent to the graph of f(x) = 2x3 5x2 + 3x 5 at x = 1.

(a) Find the equation of the tangent line to f(x) = x3 at the point where x = 2. (b) Graph the tangent line and the function on the same axes. If the tangent line is used to estimate values of the function near x = 2, will the estimates be overestimates or underestimates?

Find the equation of the line tangent to the graph of f(t) = 6t t2 at t = 4. Sketch the graph of f(t) and the tangent line on the same axes.

The time, T , in seconds for one complete oscillation of a pendulum is given by T = f(L) = 1.111L, where L is the length of the pendulum in feet. Find the following quantities, with units, and interpret in terms of the pendulum. (a) f(100) (b) f(100).

Kleibers Law states that the daily calorie requirement, C(w), of a mammal is proportional to the mammals body weight w raised to the 0.75 power.1 If body weight is measured in pounds, the constant of proportionality is approximately 42. (a) Give formulas for C(w) and C(w). (b) Find and interpret (i) C(10) and C(10) (ii) C(100) and C(100) (iii) C(1000) and C(1000)

If you are outdoors, the wind may make it feel a lot colder than the thermometer reads. You feel the windchill temperature, which, if the air temperature is 20F, is given in F by W(v) = 48.17 27.2v0.16, where v is the wind velocity in mph for 5 v 60.2 (a) If the air temperature is 20F, and the wind is blowing at 40 mph, what is the windchill temperature, to the nearest degree? (b) Find W(40), and explain what this means in terms of windchill.

(a) Use the formula for the area of a circle of radius r, A = r2, to find dA/dr. (b) The result from part (a) should look familiar. What does dA/dr represent geometrically? (c) Use the difference quotient to explain the observation you made in part (b).

Suppose W is proportional to r3. The derivative dW/dr is proportional to what power of r?

Show that for any power function f(x) = xn, we have f(1) = n.

The cost to produce q items is C(q) = 1000+2q2 dollars. Find the marginal cost of producing the 25th item. Interpret your answer in terms of costs.

The demand curve for a product is given by q = 3003p, where p is the price of the product and q is the quantity that consumers buy at this price. (a) Write the revenue as a function, R(p), of price. (b) Find R(10) and interpret your answer in terms of revenue. (c) For what prices is R(p) positive? For what prices is it negative?

A ball is dropped fromthe top of the Empire State Building. The height, y, of the ball above the ground (in feet) is given as a function of time, t, (in seconds) by y = 1250 16t2. (a) Find the velocity of the ball at time t. What is the sign of the velocity? Why is this to be expected? (b) When does the ball hit the ground, and how fast is it going at that time? Give your answer in feet per second and in miles per hour (1 ft/sec = 15/22 mph).

The yield, Y , of an apple orchard (measured in bushels of apples per acre) is a function of the amount x of fertilizer in pounds used per acre. Suppose Y = f(x) = 320+140x 10x2. (a) What is the yield if 5 pounds of fertilizer is used per acre? (b) Find f(5). Give units with your answer and interpret it in terms of apples and fertilizer. (c) Given your answer to part (b), should more or less fertilizer be used? Explain.

The demand for a product is given, for p, q 0, by p = f(q) = 50 0.03q2. (a) Find the p- and q-intercepts for this function and interpret them in terms of demand for this product. (b) Find f(20) and give units with your answer. Explain what it tells you in terms of demand. (c) Find f(20) and give units with your answer. Explain what it tells you in terms of demand.

The cost (in dollars) of producing q items is given by C(q) = 0.08q3 +75q + 1000. (a) Find the marginal cost function. (b) Find C(50) and C(50). Give units with your answers and explain what each is telling you about costs of production.

Let f(x) = x36x215x+20. Find f(x) and all values of x for which f(x) = 0. Explain the relationship between these values of x and the graph of f(x).

If the demand curve is a line, we can write p = b + mq, where p is the price of the product, q is the quantity sold at that price, and b and m are constants. (a) Write the revenue as a function of quantity sold. (b) Find the marginal revenue function.