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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 31

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 31. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | 8.1 Local Linearity and the Derivative 281 We ll use the rate at which water is decreasing today to approximate the change in water level over the next two days. dW AW - A 1 A so dW AW -- ----- for At small. dA dW AW -----At d If we let At 2 and use dW 0 -115 we have AW -115-2 -230. This makes sense the water is decreasing at a rate of 115 gallons day so after two days we d expect it to decrease by about 230 gallons. W 2 G - 230 Because W t is concave up we expect the actual water level to be a bit higher than this. EXAMPLE 8.2 Approximate 6.8. Use a flrst derivative to get a good approximation. SOLUTION Let s begin by sketching x and getting an off-the-cuff approximation of 6.8. This will help us see how a tangent line can be of use. We see that V 16.8 is a bit larger than 4. Figure 8.3 Question How do we know this Answer 16.8 is close to 16 and 6 4. Question How do we know V 16.8 is a tad more than 4 Answer We know that x is increasing between 16 and 16.8. Question How can we estimate how much to add to 4 to get a good approximation of 68 282 CHAPTER 8 Fruits of Our Labor Derivatives and Local Linearity Revisited Answer This depends on the rate at which x is increasing near x 16. This is where the derivative comes into the picture. The derivative of x at x 16 gives the rate of increase. d dx 1 _ 1 _ 1 x 16 2 16 8 Here we can adopt one of two equivalent viewpoints. Viewpoint i The Derivative as a Tool for Adjustment. We begin with an off-the-cuff approximation of V16.8 grounded in a nearby value of x that we know with certainty. Knowing the rate of increase of x at x 16 enables us to judge how to adjust our approximation in this case 4 to fit a nearby value of x. Ay for Ax small because x limAx 0 A . In this example Ax 16.8 - 16 0.8. slope 16 16 8 dy Ay dx A x Figure 8.4 dy 1 Ay 0 8 1 se-------- Ay se dx 8 0 8 8 10 So Ay 0.1 and we have the approximation V16.8 4 0.1 4.1. In other words the new y-value is approximately the old y-value 4 plus Ay where we approximate