Tangent to a Graph Problem: If you worked correctly, you found that the instantaneous

Bungee Problem: Lee Per attaches himself to a strong bungee cord and jumps off a bridge. At time t = 3 s, the cord first becomes taut. From that time on, Lees distance, d, in feet, from the river below the bridge is given by the equation d = 90 80 sin [1.2 (t 3)] a. How far is Lee from the water when t = 4? b. Find the average rate of change of d with respect to t for the interval t = 3.9 to t = 4, and for the interval t = 4 to t = 4.1. Approximately what is the instantaneous rate of change at t = 4? Is Lee going up or going down at time t = 4? Explain. c. Estimate the instantaneous rate of change of d with respect to t when t = 5. d. Is Lee going up or down when t = 5? How fast is he going? e. Which concept of calculus is the instantaneous rate of change?

a. What is the physical meaning of the derivative of a function? What is the graphical meaning? b. For the function in Figure 1-6a, explain how f(x) is changing (increasing or decreasing, quickly or slowly) when x equals 4, 1, 3, and 5. 4 1 3 5 x f(x) Figure 1-6a c. If f(x) = 5x, find the average rate of change of f(x) from x = 2 to x = 2.1, from x = 2 to x = 2.01, and from x = 2 to x = 2.001. How close are these average rates to the instantaneous rate, 40.235947...? Do the average rates seem to be approaching this instantaneous rate as the second value of x approaches 2? Which concept of calculus is the instantaneous rate? Which concept of calculus is used to find the instantaneous rate? d. Mary Thon runs 200 m in 26 s! Her distance, d, in meters from the start at various times t, in seconds, is given in the table. Estimate her instantaneous velocity in m/s when t = 2, t = 18, and t = 24. For which time intervals did her velocity stay relatively constant? Why is the velocity at t = 24 reasonable in relation to the velocities at other times? t (s) d (m) t (s) d (m) 0 0 14 89 2 7 16 103 4 13 18 119 6 33 20 138 8 47 22 154 10 61 24 176 12 75 26 200

Izzy Sinkin winds up his toy boat and lets it run on the pond. Its velocity is given by v(t) = (2t)(0.8t ) as shown in Figure 1-6b. Find, approximately, the distance the boat travels between t = 2 and t = 10. Which concept of calculus is used to find this distance? 2 4 6 8 10 12 1 2 3 4 t (s) v(t) (ft/s) Figure 1-6b

The graph in Figure 1-6c shows f(x) = 0.5x2 + 1.8x + 4 a. Plot the graph of f. Sketch your results. Does your graph agree with Figure 1-6c? f(x) 1 4 Figure 1-6c b. Estimate the definite integral of f(x) with respect to x from x = 1 to x = 4 by counting squares. c. Estimate the integral in part b by drawing trapezoids each 0.5 unit of x and summing their areas. Does the trapezoidal sum overestimate the integral or underestimate it? How can you tell? d. Use your trapezoidal rule program to estimate the integral using 50 increments and 100 increments. How close do T50 and T100 come to 15, the exact value of the integral? Do the trapezoidal sums seem to be getting closer to 15 as the number of increments increases? Which concept of calculus is used to determine this?

In Section 1-5, you started a calculus journal. In what ways do you think keeping this journal will help you? How can you use the completed journal at the end of the course? What is your responsibility throughout the coming year to ensure that keeping the journal will be a worthwhile project?

Exact Value of a Derivative Problem: You have been calculating approximate values of derivatives by finding the change in y for a given change in x, then dividing. In this problem you will apply the concept of limit to the concept of derivative to find the exact value of a derivative. Let y = f(x) = x2 7x + 11. a. Find f(3). b. Suppose that x is slightly different from 3. Find an expression in terms of x for the amount by which y changes, f(x) f(3). c. Divide the answer to part b by x 3 to get an expression for the approximate rate of change of y. Simplify. d. Find the limit of the fraction in part c as x approaches 3. The answer is the exact rate of change at x = 3.

Tangent to a Graph Problem: If you worked Problem C1 correctly, you found that the instantaneous rate of change of f(x) at x = 3 is exactly 1 y-unit per x-unit. Plot the graph of function f. On the same screen, plot a line through the point (3, f(3)) with slope 1. What do you notice about the line and the curve as you zoom in on the point (3, f(3))?

Formal Definition of Limit Problem: In Chapter 2, you will learn that the formal definition of limit is L = limxc f(x) if and only if for any positive number epsilon, no matter how small there is a positive number delta such that if x is within delta units of c, but not equal to c, then f(x) is within epsilon units of L. Notes: limxc f(x) is read the limit of f(x) as x approaches c. Epsilon is the Greek lowercase letter . Delta is the Greek lowercase letter . 123456 1 2 3 4 5 6 x (s) f(x) (ft) Figure 1-6d Figure 1-6d shows the graph of the average velocity in ft/s for a moving object from 3 s to x s given by the function f(x) = 4x2 19x + 21 x 3 From the graph you can see that 5 is the limit of f(x) as x approaches 3 (the instantaneous velocity at x = 3), but that r (3) is undefined because of division by zero. a. Show that the (x 3) in the denominator can be canceled out by first factoring the numerator, and that 5 is the value of the simplified expression when x = 3. b. If  = 0.8 unit, on a copy of Figure 1-6d show the range of permissible values of f(x) and the corresponding interval of x-values that will keep f(x) within 0.8 unit of 5. c. Calculate the value of to the right of 3 in part b by substituting 3 + for x and 5.8 for f(x), then solving for . Show that you get the same value of to the left of 3 by substituting 3 for x and 4.2 for f(x). d. Suppose you must keep f(x) within  units of 5, but you havent been told the value of . Substitute 3 + for x and 5 +  for f(x). Solve for in terms of . Is it true that there is a positive value of for each positive value of , no matter how small, as required by the definition of limit? e. In this problem, what are the values of L and c in the definition of limit? What is the reason for the clause . . . but not equal to c in the definition?