2.7: Derivatives and Rates of Change
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1 2.7: Derivatives and Rates of Change Recall from section 2.1, that the tangent line to a curve at a point x = a has a slope that can be found by finding the slopes of secant lines through the curve at x = a and some second point x = b, then taking the limit of the values you find as b gets closer to a. We can write this as f(x) f(a) m = lim CDE x a If I substitute "a plus a little" in for x, my equation for slope becomes m = lim (ENO) E f(a + h) f(a) (a + h) a and from there, f(a + h) f(a) m = lim
2 With this new formula, we can find the slope of a line tangent to a curve as long as we know the function (and as long as the limit we need to take exists). Example: Find the slope of the line tangent to the curve f(x) = x X at x = 1. Since m = lim, we need to find f(a + h) and f(a) and [(ENO)D[(E) O S O "plug them in" to our formula. f(a) = a X f(a + h) = (a + h) X = a X + 2ah + h X therefore, f(a + h) f(a) (a X +2ah + h X ) a X 2ah + h X lim = lim = lim h = lim 2a + h = 2a 1 Now, we wanted the slope where x = 1, so we replace our "variable" x-value a with 1, and we have that m = 2(1) = 2 when x = 1. Example: Use the slope of the line tangent to the curve f(x) = x X at x = 1 to write the equation of the line. Since the equation of a line can be written y = mx + b and we know m and we have a point (x, f(x)), we can find the equation.
3 y = mx + b f(1) = 2(1) + b 1 X = 2 + b 1 2 = b b = 1 Thus, y = 2x 1 is the equation of the line we were asked for. Derivatives: The process we have just endured is so common and so useful, that we give it a special name, and we will use it frequently. It is also possible to use the formula instead, although it is much more common to use the formula found in Definition 4. Rates of Change: The derivative is a powerful concept. It represents the slope of the line tangent to a curve at the point on the curve associated with x = a, but what is a slope? "Rise over run"? Sure, but what does that mean? It means the change in the dependent variable (y) with respect to the change in the independent variable (x).
4 It means the change in position (of a car, a particle, a projectile, a person) for each additional minute (hour, second, etc.) of time that passes. It means the difference in cost to send a package for each additional ounce in weight. It means the change in a city's population for each additional year that passes. It means the change in the amount of money your bank account for each additional month you leave funds in the account. In each case, slope means the amount of change recorded for "y" for every unit of change in "x". Slope is a rate of change in many contexts. So the derivative is a rate of change in many contexts. We know the slope as a way of determining an average rate of change from Math 97 and 98: m = f (x X ) f(x i ) x X x i As a rate of change at a particular instant in time, we use the definition of a derivative (the limit of the slope of secant lines as x approaches a) to describe the instantaneous rate of change. The derivative is the slope of a tangent line, so since the slope is a rate of change, the derivative is a rate of change. The steeper the slope of a tangent at a given point on a curve, bigger the derivative is, and the faster the "y" values are changing for every increase in one "x" value. If the slope of a tangent line is shallow, the derivative will be small because the y values are not changing very fast.
5 Example: The graph shows the influence of the temperature T on the maximum sustainable swimming speed S of Coho salmon. (a) What is the meaning of the derivative S (T)? What are the units? (b) Estimate the values of S (15) and S (25) and interpret them. (a) The derivative of the function S(T) (S (T)) is the rate of change of the speed of the salmon at a particular temperature, T. Or, when the temperature is C degrees Celsius, the salmon's maximum speed changes by S (T). (b) The derivative, or the slope, is the change in speed in cm/s for every ("per") degree Celsius, so the units are rs/t, or centimeters per v second per degree. The rate at which the salmon can swim changes for each additional degree. Homework: 5-8, 11, 12, 15, 17-20, 27-37, 53, 54
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