By George Kempf

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Find the derivative of f(ln x). Solution: Let y = f(1nx) and U = lnx. Then y = f(u), U = Inx. 9 du dx X 2 Find dyldx and express it solely in terms of x: y=fi, u = u3 + I , v=sinx. 10 du dv dx 2 J S r n ’ Figure 2-26 contains the graph of a function given by the equation f ( x ) = ux3 bx c. What are the values of a , b, and c? + + Solution: First note that f ( 0 ) = 0. Since f(0) = c , we see that c = 0, thereby reducing f ( x ) to ax3 bx. We need 2 additional conditions to determine a and b, and we get them from the fact that f(1) = -1 and f’(1) = 0.

We can use the second derivative to determine the concavity of the function. 1 Another fact to take note of is that not every function is given by a formula. Situations exist in which the graph of a function is presented, and we must then answer questions about its behavior and that of its derivative. We will see problems of this type in this section. Find the slope of the curve y = 2x3 at the point P = (4,128), directly from the definition. Solution: If we choose Q = (5,250), then the slope S of the secant joining P and Q is S= 250 - 128 = 122.

B. C. ) and the vertical axis represents the number of hours of daylight in your city at the corresponding time. ) Do this for a 2-year period. At what time of the year are the days longest? At what time of the year is the length of the day (that is, the number of hours of daylight) increasing at the most rapid rate? 35 Figure 2-32 is the graph of a function f . 36 a. Sketch the graph of f’. b. Sketch the graph of a second function, g, which satisfies g ( 0 ) = 0, g’(x) = f’(x) for all x in [0,5].

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Abelian integrals by George Kempf


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