Math 251 Lecture Notes
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1 Lecture Notes 2.1: Average Rates of Change Example 1. A population of bees was happily residing in someone s backyard a few years ago. Unfortunately, the population died off before the year s end. Let B(t) be the number of bees (in thousands of bees) t months after July 1, We will model this population by B(t) = 32 2t 2. The bees did not die off at a constant rate (as the function B is not linear), but we can still determine the average rate of change at which they died off. (a) From the time the bees hit their peak (when t = 0) to when they had officially died off, what was the average rate of change in the size of the bee population? Write a sentence interpreting this rate, making sure to include appropriate units. 1
2 The average rate of change of a continuous function f over the interval [a, b] is given by f(b) f(a) b a (b) Write and fully simplify the formula for the average rate of change in B(t) = 32 2t 2 over the interval [t 0, t 1 ]. Instructor: A.E.Cary Page 2 of 8
3 The difference quotient of a function defined by y = f(x) is the expression f(x + h) f(x) h **This formula can be applied to any function, regardless of the function name or independent variable, but must be adapted appropriately.** (c) Find and fully simplify the difference quotient for the function defined by B(t) = 32 2t 2. Instructor: A.E.Cary Page 3 of 8
4 (d) Let s explore how the average rate of change and the difference quotient are related. We ll do this by finding the average rate of change in B over the interval [t, t + h]. (e) Use the forumlas found for the average rate of change AND the difference quotient to determine the average rate of change of B over the intervals [1, 3] and [3, 3.5]. Instructor: A.E.Cary Page 4 of 8
5 What if we wanted to know the rate of change that was occurring exactly when t = 3? (f) One approach is to use the average rate of change formula with 3 and t 1. Here, we will approximate the exact rate of change when t = 3 by adjusting the values of t 1, as shown in Table 1 below. Table 1. Average Rate of Change t 0 t 1 Average Rate of Change (g) A second approach is to use the formula we found for the difference quotient. Here, we will approximate the exact rate of change when t = 3 by adjusting the values of h, as shown in Table 2 below. Table 2. Difference Quotient t h Difference Quotient Instructor: A.E.Cary Page 5 of 8
6 Example 2. The graph of y = B(t) and the secant line from (1, B(1)) to another point (1+h, B(1+h)) is shown below. Use this to answer the following. Figure 1 y t 4 (a) The run between the two points is or. (b) The rise between the two points is or. (c) The slope of the secant line is or. Instructor: A.E.Cary Page 6 of 8
7 Example 3. Find the difference quotient for the function defined by g(x) = 4. Use this to x + 3 determine the slope of the secant line from x = 2 to x = 1. Figure 2. Graph of y = g(x) y 6 4 y = x 4 x = -3 6 Instructor: A.E.Cary Page 7 of 8
8 Example 4. A spring is stretched and oscillates from its original position until eventually coming to rest. Let p(t) be the displacement of the spring (in cm) from its original position t seconds after being stretched. This will be modeled by p(t) = 10 ( 1 2) t cos (πt). (a) Find and interpret p(0.75) p(0.25). Round accurately to two decimal places. 0.75s 0.25s (b) Using your graphing calculator, find the slope of the secant line from t = 1 to t = 0.9, 0.99, 0.999, 1.001, 1.01, and 1.1 for the function p. Use this to estimate the slope of the tangent line at t = 1. Instructor: A.E.Cary Page 8 of 8
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Chapter 4 Nonlinear equations 4.1 Root finding Consider the problem of solving any nonlinear relation g(x) = h(x) in the real variable x. We rephrase this problem as one of finding the zero (root) of a