CK-12 Foundation's CK-12 Calculus, Volume 2 PDF

By CK-12 Foundation

ISBN-10: 1935983172

ISBN-13: 9781935983170

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Extra info for CK-12 Calculus, Volume 2

Example text

If is the length of the sides of any arbitrary square, then, by similar triangles (Figure 7b), Since the cross-sectional area at is Using the volume formula, Using substitution to integrate, we eventually get Therefore the volume of the pyramid is , which agrees with the standard formula. 12 Suppose a function is continuous and non-negative on the interval and suppose that is the region between the curve and the axis (Figure 8a). If this region is revolved about the axis, it will generate a solid that will have circular cross-sections (Figure 8b) with radii of at each Each cross-sectional area can be calculated by Since the volume is defined as the volume of the solid is Volumes by the Method of Disks (revolution about the axis) Because the shapes of the cross-sections are circular or look like the shapes of disks, the application of this method is commonly known as the method of disks.

Function does not have an inverse. Function does not have an inverse. on which is negative on the interval in question, so is monotonically decreasing. Exponential and Logarithmic Functions Learning Objectives A student will be able to: Understand and use the basic definitions of exponential and logarithmic functions and how they are related algebraically. Distinguish between an exponential and logarithmic functions graphically. A Quick Algebraic Review of Exponential and Logarithmic Functions Exponential Functions Recall from algebra that an exponential function is a function that has a constant base and a variable exponent.

To find a formula for this inverse, we start with the exponential function Interchanging and Projecting the logarithm to the base on both sides, Thus is the inverse of This implies that the graphs of and are reflections of one another about the line The figure below shows this relationship. Similarly, in the special case when the base the two equations above take the forms and The graph below shows this relationship: Before we move to the calculus of exponential and logarithmic functions, here is a summary of the two important relationships that we have just discussed: The function is equivalent to if and The function is equivalent to if and You should also recall the following important properties about logarithms: To express a logarithm with base in terms of the natural logarithm: To express a logarithm with base in terms of another base : Review Questions Solve for Review Answers and Differentiation and Integration of Logarithmic and Exponential Functions Learning Objectives A student will be able to: Understand and use the rules of differentiation of logarithmic and exponential functions.

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CK-12 Calculus, Volume 2 by CK-12 Foundation


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