Homework, Page 148 Use the Product Rule to find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 148 Use the Product Rule to find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 148 Use the Quotient Rule to find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 148 Calculate the derivative in two ways. First use the Product or Quotient Rule, then rewrite and apply the Power Rule. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 148 Calculate the derivative using the appropriate rule(s). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 148 Calculate the derivative using the appropriate rule(s). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 148 Calculate the derivative using the appropriate rule(s). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 148 Calculate the derivative using the appropriate rule(s). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 148 Calculate the derivative using the appropriate rule(s). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 148 Calculate the derivative using the appropriate rule(s). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 148 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 148 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Chapter 3: Differentiation Section 3.4: Rates of Change Jon Rogawski Calculus, ET First Edition Chapter 3: Differentiation Section 3.4: Rates of Change Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

The average rate of change (ROC) of a function over an interval is: The instantaneous ROC of a function at a point is: We must realize that the average ROC of a function is the slope of the secant line and the instantaneous ROC is the slope of the tangent line. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Figure 1 illustrates measuring the average ROC of f (x) over (x0, x1), while Figure 2 illustrates measuring the instantaneous ROC at x = x0. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 158 2. Find the ROC of the volume of a cube with respect to the length of its side s when s = 3 and s = 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 158 2. Find the ROC of the volume of a cube with respect to the length of its side s when s = 3 and s = 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Table 1 illustrates some of the data used to construct the graph in Figure 3. The slope of the secant line in Figure 3 gives the average ROC of the temperature from midnight to 12:28PM on July 6, 1997. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

slope of the tangent to the area curve increases. Notice that as r increases, the area of the blue band increases, and the slope of the tangent to the area curve increases. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 158 8. Calculate the ROC dA/dD, where A is the surface area of a sphere of diameter D. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 158 8. Calculate the ROC dA/dD, where A is the surface area of a sphere of diameter D. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Marginal cost, in economics, gives the cost of delivering one more unit. Figure 5 shows how the cost of a flight is less per passenger, the more passengers there are. Thus, the marginal cost of flying each additional passenger decreases as the number of passengers increases. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

From the graph, when is the car moving toward its destination? When is the car standing still? When is the car going towards is origin? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

From the graph, when is the car moving toward its destination? When is the car standing still? When is the car going towards is origin? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

(b) Is the truck speeding up or slowing down? A truck enters a freeway off-ramp at t = 0. Its position after t seconds is s(t) = 84t – t3 ft for 0 ≤ t ≤ 5. (a) How fast is the truck traveling at the moment it enters the off-ramp? (b) Is the truck speeding up or slowing down? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

(b) Is the truck speeding up or slowing down? A truck enters a freeway off-ramp at t = 0. Its position after t seconds is s(t) = 84t – t3 ft for 0 ≤ t ≤ 5. (a) How fast is the truck traveling at the moment it enters the off-ramp? (b) Is the truck speeding up or slowing down? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Galileo used a device such as in Figure 9 to investigate the motion of objects moving solely under the influence of gravity. From his experiments, he deduced that, in the absence of air resistance, the velocity of a falling object is proportional to the time it has been falling. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

The equations below may be used to determine the vertical position and velocity of an object, absent air resistance, under the influence of gravity alone. For the equations, s0 is initial height, v0 is initial velocity, and t is time. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example A baseball is tossed vertically upward with an initial velocity of 85 ft/s from the top of a 65-ft high building. a. What is the height of the ball after 0.25 s? b. Find the velocity of the ball after 1 s. c. When does the ball hit the ground? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example A baseball is tossed vertically upward with an initial velocity of 85 ft/s from the top of a 65-ft high building. a. What is the height of the ball after 0.25 s? b. Find the velocity of the ball after 1 s. c. When does the ball hit the ground? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 158 18. The escape velocity at a distance r meters from the center of the Earth is vesc = (2.82 x 107)r –½ m/s. Assuming the radius of the Earth is 6.77 x 106 m, calculate the rate at which vesc changes with respect to distance at the Earth’s surface. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 158 18. The escape velocity at a distance r meters from the center of the Earth is vesc = (2.82 x 107)r –½ m/s. Assuming the radius of the Earth is 6.77 x 106 m, calculate the rate at which vesc changes with respect to distance at the Earth’s surface. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

WS 3.4.pdf Rogawski Calculus Copyright © 2008 W. H. Freeman and Company