If M is the mass of the earth and G is a constant, the acceleration due to gravity, g

Let f(x)=7. Using the definition of the derivative, show that f (x)=0 for all values of x. 2. Let f(x) = 17x+11. Use the definition of

Let f(x) = 17x+11. Use the definition of the derivative to calculate f (x).

For Exercises 35, determine if the derivative rules from this section apply. If they do, find the derivative. If they dont apply, indicate why.

For Exercises 35, determine if the derivative rules from this section apply. If they do, find the derivative. If they dont apply, indicate why.

For Exercises 35, determine if the derivative rules from this section apply. If they do, find the derivative. If they dont apply, indicate why.

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Exercises 649, find the derivatives of the given functions. Assume that a, b, c, and k are constants

For Problems 5055, determine if the derivative rules from this section apply. If they do, find the derivative. If they dont apply, indicate why

For Problems 5055, determine if the derivative rules from this section apply. If they do, find the derivative. If they dont apply, indicate why

For Problems 5055, determine if the derivative rules from this section apply. If they do, find the derivative. If they dont apply, indicate why

For Problems 5055, determine if the derivative rules from this section apply. If they do, find the derivative. If they dont apply, indicate why

For Problems 5055, determine if the derivative rules from this section apply. If they do, find the derivative. If they dont apply, indicate why

For Problems 5055, determine if the derivative rules from this section apply. If they do, find the derivative. If they dont apply, indicate why

The graph of y = x3 9x2 16x + 1 has a slope of 5 at two points. Find the coordinates of the points

Find the equation of the line tangent to the graph of f at (1, 1), where f is given by f(x)=2x3 2x2 + 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, will the estimates be overestimates or underestimates

Find the equation of the line tangent to y = x2 + 3x 5 at x = 2.

Find the equation of the line tangent to f(x) at x = 2, if f(x) = x3 2 4 3x.

Using a graph to help you, find the equations of all lines through the origin tangent to the parabola y = x2 2x + 4. Sketch the lines on the graph

On what intervals is the graph of f(x) = x4 4x3 both decreasing and concave up?

For what values of x is the function y = x5 5x both increasing and concave up?

If f(x)=4x3 + 6x2 23x + 7, find the intervals on which f (x) 1

If f(x) = (3x + 8)(2x 5), find f (x) and f(x).

Given p(x) = xn x, find the intervals over which p is a decreasing function when: (a) n = 2 (b) n = 1 2 (c) n = 1

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

The height of a sand dune (in centimeters) is represented by f(t) = 700 3t 2, 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.

A ball is dropped from the top of the Empire State building to the ground below. The height, y, of the ball above the ground (in feet) is given as a function of time, t, (in seconds) by y = 1250 16t 2 . (a) Find the velocity of the ball at time t. What is the sign of the velocity? Why is this to be expected? (b) Show that the acceleration of the ball is a constant. What are the value and sign of this constant? (c) 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)

At a time t seconds after it is thrown up in the air, a tomato is at a height of f(t) = 4.9t 2 + 25t+ 3 meters. (a) What is the average velocity of the tomato during the first 2 seconds? Give units. (b) Find (exactly) the instantaneous velocity of the tomato at t = 2. Give units. (c) What is the acceleration at t = 2? (d) How high does the tomato go? (e) How long is the tomato in the air?

A particle is moving on the x-axis, where x is in centimeters. Its velocity, v, in cm/sec, when it is at the point with coordinate x is given by v = x2 + 3x 2. Find the acceleration of the particle when it is at the point x = 2. Give units in your answer.

Let f(t) and g(t) give, respectively, the amount of water (in acre-feet) in two different reservoirs on day t. Suppose that f(0) = 2000, g(0) = 1500 and that f (0) = 11, g (0) = 13.5. Let h(t) = f(t) g(t). (a) Evaluate h(0) and h (0). What do these quantities tell you about the reservoir? (b) Assume h is constant for 0 t 250. Does h have any zeros? What does this tell you about the reservoirs?

If M is the mass of the earth and G is a constant, the acceleration due to gravity, g, at a distance r from the center of the earth is given by g = GM r2 . (a) Find dg/dr. (b) What is the practical interpretation (in terms of acceleration) of dg/dr? Why would you expect it to be negative? (c) You are told that M = 61024 and G = 6.671020 where M is in kilograms and r in kilometers. What is the value of dg/dr at the surface of the earth (r = 6400 km)? (d) What does this tell you about whether or not it is reasonable to assume g is constant near the surface of the earth?

The period, T , of a pendulum is given in terms of its length, l, by T = 2 l g

(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).

What is the formula for V , the volume of a sphere of radius r? Find dV /dr. What is the geometrical meaning of dV /dr?

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

Given a power function of the form f(x) = axn, with f (2) = 3 and f (4) = 24, find n and a.

Is there a value of n which makes y = xn a solution to the equation 13x(dy/dx) = y? If so, what value?

Using the definition of derivative, justify the formula d(xn)/dx = nxn1 . (a) For n = 1; for n = 3. (b) For any negative integer n.

(a) Find the value of a making f(x) continuous at x = 1: f(x) = ax 0 x 1 x2 + 3 1 < x 2. (b) With the value of a you found in part (a), does f(x) have a derivative at every point in 0 x 2? Explain.

Find values of a and b making f(x) continuous and differentiable on 0 <x< 2: f(x) = ax + b 0 x 1 x2 +3 1 < x 2.

In Problems 8384, explain what is wrong with the statement.

In Problems 8384, explain what is wrong with the statement.

In Problems 8587, give an example of:

In Problems 8587, give an example of:

In Problems 8587, give an example of:

Are the statements in Problems 8890 true or false? Give an explanation for your answer.

Are the statements in Problems 8890 true or false? Give an explanation for your answer.

Are the statements in Problems 8890 true or false? Give an explanation for your answer.

Suppose that f and g exist and that f and g are concave up for all x. Are the statements in Problems 9192 true or false for all such f and g? If a statement is true, explain how you know. If a statement is false, give a counterexample.

Suppose that f and g exist and that f and g are concave up for all x. Are the statements in Problems 9192 true or false for all such f and g? If a statement is true, explain how you know. If a statement is false, give a counterexample.