Figure 7-8a, on the next page. In 116, you will analyze Leias motion.If you take the sum

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion. On a copy of Figure 7-8a, draw a narrow vertical strip of width dt. Show a sample point (t, v(t)) on the graph within the strip. What physical quantity does v(t) dt represent?

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.If you take the sum v(t) dt from t = 0 to t = 8, what calculus concept equals the limit of this sum as dt approaches zero? Leia figures that her velocity is given by v(t) = t3 21t 2 + 100t + 80 Use the fundamental theorem of calculus to find the distance she travels from t = 0 to t = 8

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.Calculate midpoint Riemann sums with n = 100 and n = 1000 increments. How do the results confirm that the fundamental theorem gives the correct answer for the integral even though it has nothing to do either with Riemann sums or with limits?

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.On a copy of Figure 7-8a, draw a representation of an upper sum with n = 8 increments.

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.Explain why, for an integrable function, any Riemann sum is squeezed to the same limit as the upper and lower sums as the widths of the increments approach zero.

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.Write the definition of definite integral. State the fundamental theorem of calculus. Be sure to tell which is which.

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.Calculate the integral in Problem 3 numerically by using your graphers integrate feature. Calculate the integral again graphically by counting squares. Compare the answers with the exact value.

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion. Use symmetric difference quotients with t = 0.1 min and t = 0.01 min to estimate the rate of change of Leias velocity when t = 4 min.

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.Use symmetric difference quotients with t = 0.1 min and t = 0.01 min to estimate the rate of change of Leias velocity when t = 4 min.

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.Write the definition of derivative

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.For most types of functions there is a way to find the derivative algebraically. Use the appropriate method to find the exact rate of change of Leias velocity when t = 4.

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.At t = 4, was Leia speeding up or slowing down? Justify your answer.

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.On a copy of Figure 7-8a, draw a line at the point (4, v(4)) with slope (4). Show how you constructed the line. How is the line related to the graph?

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.What is the physical name of the instantaneous rate of change of velocity?

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.Leias maximum velocity seems to occur at t = 3. Use derivatives appropriately to find out whether the maximum occurs when t is exactly 3 s.

Figure 7-8a, on the next page. In Problems 116, you will analyze Leias motion.Find an equation for (t), the second derivative of v(t) with respect to t. Related Rates Problems: Suppose that you have constructed the diagram shown in Figure 7-8b on a computer screen using a dynamic geometry utility such as The Geometers Sketchpad. The curve has equation y = 6e0.5x. A vertical line y units long connects the x-axis to a point on the graph, and a diagonal line z units long connects this point on the graph to the origin. Figure 7-8b

Suppose that you drag the point on the x-axis at a constant rate of 0.3 unit per second. Find dy/dt when x = 2. Is y increasing or decreasing at this instant? At what rate?

Find dz/dt when x = 2. Is z increasing or decreasing at this instant? At what rate? Compound Interest Problems: When money is left in a savings account that compounds interest continuously, the instantaneous rate at which the money increases, dm/dt, is directly proportional to m, the amount in the account at that instant.

Write a differential equation that expresses this property.

Solve the differential equation in Problem 19 for m as a function of t, and show the steps that you use

In one word, how does m vary with t?

The solution in Problem 20 is called the ? solution of the differential equation.

Find the particular solution in Problem 20 if m is $10,000 at t = 0 and $10,900 at t = 1.

. In Problem 23, the amount of money in the account grew by $900 in one year. True or false: The amount of money will grow by $9000 in 10 yr. Justify your answer. Discrete Data Problems: The techniques of calculus were invented for dealing with continuous functions. These techniques can also be applied to functions specified by a table of data. This table gives values of y for various values of x. x y 30 74 32 77 34 83 36 88 38 90 40 91 42 89

Use Simpsons rule to estimate y dx.

Estimate dy/dx if x = 36. Show how you got your answer. Mean Value Theorem Problems: The proof of the fundamental theorem is based on the mean value theorem. This theorem is a corollary of Rolles theorem.

State Rolles theorem.

Sketch a graph that illustrates the conclusion of the mean value theorem.

Figure 7-8c shows the graph of function f. On a copy of this figure, sketch the derivative graph, . Figure 7-8c

The function has a discontinuity at x = 1. Sketch the graph. What type of discontinuity is it?

The function g(x) = x1/3(x 1) has g(0) = 0. Show that (0) is undefined. Show what the graph of g looks like in a neighborhood of x = 0. You may use your graphers cube root function, if it has one. Area and Volume Problems: Figure 7-8d shows the graphs of y = e0.2x and y = 0.6x. Figure 7-8d

Find numerically the two intersections of the graphs shown in the figure. Store these, without rounding. Find the area of the narrow region bounded by these two graphs between the two intersections.

The region bounded by the two graphs in Problem 32 from x = 0 to the first intersection is rotated about the x-axis to form a solid. Find the volume of this solid. Differential Equation Problems: Figure 7-8e shows the slope field for the differential equation Figure 7-8e

On a copy of Figure 7-8e, sketch the particular solutions that contain the points (0, 3) and (10, 4).

The two solutions in Problem 34 share a common asymptote. Sketch the asymptote. State an initial condition that would give the asymptote as the graph of the solution.

Solve the differential equation by separating the variables and integrating. Find the equation of the particular solution that contains the point (10, 4).

Use the function in Problem 36 to calculate the exact value of y when x = 10.5.

Demonstrate that you understand the idea behind Eulers method by calculating the first point to the right of the point (10, 4) in Problem 36, with x = 0.5. How does this value compare with the exact value in Problem 37?

Find: (sin1 x3 )

Find if x = ln (cos t) and y = sec t.

Find:

Find (x) if h(x) = 5x

Find:

Plot the graph of the fraction given in Problem 43. Sketch the result. Show how the graph confirms your answer to Problem 43.

Write what you think is the one most important thing you have learned so far as a result of taking calculus.

Write one thing in calculus that you are still not sure about.