How To Study with DWFK Precalculus

Chapter 10: An Introduction to Calculus: Limits, Derivatives, and Integrals

We have made some significant changes to this chapter from previous editions of the book. Our goal is to set the stage for the study of calculus by looking at two classical problems of motion. Through the tools of analytic geometry (which should be well-honed by the end of this course), we show that one problem leads geometrically to the slopes of tangent lines, while the other leads geometrically to areas under curves. We show how both problems can be solved using limits (intuitively defined), and we sprinkle the whole chapter liberally with perspectives from history.

In short, we hope that this chapter will give you an appreciation for the "calculus step" that you will make next year. We want to set the stage for you without spoiling the first act.

Section 10.1 Section 10.3

Section 10.2 Section 10.4

Section 10.1 Limits and Motion: The Tangent Problem

Objectives
You will understand the connection between instantaneous velocity and the slope of the tangent line to the position curve. Using the definition of the derivative, you will be able to compute instantaneous velocities in simple cases.

Key Ideas

Average rate of change Differentiable

Average velocity Instantaneous velocity

Derivative at a point Leibniz notation

Derivative of a function Limit at a

Study Tips
We have written this chapter with the understanding that we are not beginning a calculus course; we are ending a precalculus course. The objective here is to motivate the derivative as a solution to a particular algebraic problem, not to teach you everything you ever need to know about derivatives.

We urge you to read this section and work through the examples and explorations carefully.

Technology Tips
We have moved the numerical derivative to the same section as the numerical integral (Section 10.4) in order to keep attention in this section focused simply on the Tangent Line Problem. That leaves this section pretty much technology-free, which is probably best for getting the proper message across.
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Section 10.2 Limits and Motion: The Area Problem

Objectives
You will understand the connection between distance traveled and the area under the velocity curve. They will be able to compute definite integrals in simple cases by computing areas.

Key Ideas

Definite integral Limit at infinity

Distance traveled Rectangular approximation method (RAM)

Integrable on [a, b] Riemann sum

Study Tips
Keep our objectives in mind when working through this section: We want to motivate the definite integral as a solution to a particular algebraic problem, not to teach you everything you ever need to know about definite integrals.

In particular, we hope that your instructor will not mention "antiderivatives" in this or any other section of the chapter. We are just setting the stage, and suggesting that the area problem can be solved by antiderivatives really is spoiling the plot of next year's drama.

As in the previous Study Tip, we urge you to read this section and work through the examples and explorations carefully.

Technology Tips
There are some nice programs out there that will compute LRAM and RRAM sums and even draw the rectangles under the curve. A list of selected programs and the web sites from which they can be obtained is available elsewhere on this web site. Shown below is one such program, computing a RRAM sum using 20 rectangles on the interval [0, ]:

You will get a chance to write your own numerical integrator on your home screen when you work through Exploration 1 on page 832.
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Section 10.3 More On Limits

Objectives
You will have an intuitive understanding of one-sided limits, two-sided limits, and limits involving infinity. You will be able to use the properties of limits to evaluate limits of these types.

Key Ideas

Infinite limit One-sided limit

Left-hand limit Properties of limits

Limit at a Right-hand limit

Limit at infinity Two-sided limit

Study Tips
This section is included so that instructors who want to give a complete treatment of limits will be able to do so. The "stage-setting" objectives of this chapter will have been met in Sections 10.1 and 10.2, so this deeper treatment of limits really does get into the beginnings of a calculus course. We motivate the material by tying it historically to the development of calculus. Examples and exercises continue to reinforce the links between the algebra and the geometry.

Technology Tips
Calculators are good for investigating limits, but it is good to keep stressing the fact that such explorations are not the same as proofs. Most technology is unable to distinguish a limit of 2, for example, from a limit of 2.00000000000000000000000000000000001, whereas algebraic techniques can.
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Section 10.4 Numerical Derivatives and Integrals

Objectives
You will be able to use calculators and numerical techniques to estimate derivatives and definite integrals of functions described algebraically or numerically (data).

Key Ideas

Numerical derivative
Numerical integral
Symmetric difference quotient

Study Tips
These numerical methods are included because they are exactly the kinds of things that you are able to do now to explore calculus. You can get the calculator to do the work while you further your understanding of what derivatives and integrals really are. This section is less important than the first two sections of the chapter for getting the big picture, but it may bring parts of that picture into sharper focus.

Technology Tips
It almost goes without saying that this is a section that relies heavily on having the technology at hand. In a sense, Section 10.4 is just a series of technology tips for Sections 10.1 and 10.2!