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AP Calculus AB Course Review

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If we plot displacement as a function of time, we see that average velocity equals the slope of the secant line.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As illustrated in Figure 4, as the end point of the secant line approaches t = 0.5, the slope of the secant line gets closer to the slope of the tangent line.

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As we have seen on the previous slides, two relationships have developed: 1.As the time interval shrinks to zero, the average velocity approaches the instantaneous velocity. 2.As the time interval shrinks to zero, the slope of the secant line approaches the slope of the tangent line. From these relationships, we may conclude: Instantaneous velocity is equal to the slope of the line tangent to the velocity curve at the time in question. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recalling that the distance between points a and b is |b – a|, If the values of f (x) do not converge to any finite value as x approaches c, we say that

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 7 shows the graph of a piecewise function. The limits as x approaches 0 and 2 do not exist, but the limit as x approaches 4 appears to exist, as the graph on both sides of x = 4 seems to converge on the same value of y. The following one-sided limits also appear to exist: but DNE as the values of f (x) continue to oscillate between ±1 as x gets ever closer to 0.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The discussion on the previous slide leads to the following definition: If a continuous function is defined on [a, b], and c is any point on (a, b) then the following conditions exist:

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company This may also be written as:

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 8 illustrates how the secant lines at a jump discontinuity fail to converge on the tangent line at one side or the other of the discontinuity. This leads to Theorem 3.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As noted above, the derivative of a product is NOT the product of the derivatives.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As noted above, the derivative of a quotient is NOT the quotient of the derivatives.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The average rate of change (ROC) of a function over an interval is: The instantaneous ROC of a function at a point is: The average ROC of a function is the slope of the secant line and the instantaneous ROC is the slope of the tangent line.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The equations below may be used to determine the vertical position and velocity of an object, absent air resistance, under the influence of gravity alone. For the equations, s 0 is initial height, v 0 is initial velocity, and t is time.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recalling that all angles will be measured in radians, unless otherwise stated, the derivatives of the sine and cosine functions are given below.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The derivatives of the other four trig functions may be found by using the definitions of the functions in terms of sine and cosine and differentiating using the quotient rule. The results are given in Theorem 2.

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Composite Functions Composite functions are combinations of simpler functions, for instance f (x) = sin e x. None of the rules of differentiation we learned thus far permit us to differentiate a composite function. Remembering that the composition of f (x) and g (x) is written as f ○ g (x) or f (g (x)), we use the Chain Rule to differentiate composites. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If we choose, we may also represent a composite function of f (x) and u (x) as f (u). The chain rule then becomes: which may also be written as:

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Combining the Chain Rule with both the Power Rule and Exponential Rule, we obtain the General Power and Exponential Rules:

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company When we learned about the graphs of functions, we learned about translations, stretches, and shrinks. If a function f (x) is differentiable, then a translated and stretched (shrunk) version of it, noted as f (kx + b), is also differentiable. Specifically,

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Implicit Differentiation

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recall that inverse functions are defined for one-to-one functions, and their graphs are reflections about the line y = x of the graph of the basic function, and for every point (a, b) on the graph of the basic function there is a corresponding point (b, a) on the graph of the inverse function. We then have Theorem 1, which states:

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company To find the slope of the graph of arcsine or arccosine, we need to first differentiate the function, as shown in Theorem 2.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Natural logarithms are frequently used in physics and engineering, and their derivative is found using Theorem 2. Remember that we can only find logarithms of positive numbers.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Two of the most important calculus facts about exponential functions are given in the following box.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Applying the Chain Rule to the derivative of the natural log, we have the following equality.

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Related Rates

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As illustrated in Figure 4, the tangent line gives a good approximation of the curve in a small neighbor of the point of tangency. This permits us to estimate values of a function proximate to the point of tangency linearization.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The extreme values of a function f (x) on an interval I are the minimum and maximum values of f (x) for If f (x) is continuous on [a, b], then f (x) has a minimum and a maximum value on [a, b]. f (c) is a local maximum if f (x) ≤ f (c) for all x in some open interval around c. Similarly, f (c) is a local minimum if f (x) ≥ f (c) for all x in some open interval around c. c is a critical point of the function f (x) if either f ' (c) = 0 or f ' (c) D.N.E.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A discontinuous function may or may not have a min or max. A function defined on an open interval may or may not have a min or max. A continuous function defined on a closed interval will have both a min and a max.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Examples of the local min and max are seen in Figure 3.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As shown in Figure 4(A), the tangent line at a min or max is horizontal. But in Figure 4(B), there is not tangent at the min, leading to the following definition:

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 9 illustrates a situation where f (6) is a maximum, but 6 is not a critical point. But f (4) is a minimum and 4 is a critical point.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Rolle’s Theorem is illustrated in Figure 13.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If a function is defined over a closed interval, then there is some point x = c on the interval such that the slope at point c equals the slope of the secant line joining the end points of the interval. This is known as the Mean Value Theorem. Theorem 1 is illustrated in Figure 1.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A function f (x) is monotonic if it is strictly increasing or strictly decreasing on some interval (a, b).

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Not stated, but implied in Theorem 3 is that the sign of f ′(x) may change at a critical point, but it may not change anywhere in the interval between two consecutive critical points.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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