ELEM CALC & ITS APPLICS
ELEM CALC & ITS APPLICS MA 123
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This 7 page Class Notes was uploaded by Kennith Herman on Friday October 23, 2015. The Class Notes belongs to MA 123 at University of Kentucky taught by David Little in Fall. Since its upload, it has received 19 views. For similar materials see /class/228130/ma-123-university-of-kentucky in Mathematics (M) at University of Kentucky.
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Date Created: 10/23/15
MA123 Chapter 4 Computing some derivatives pp 6982 Date Chapter Goals 0 Use the de nition to calculate some derivatives 0 Use the de nition to approximate some derivatives 0 Investigate further the notions of continuity and di erentiability Assignment 06 Due date Feb 9 Assignment 07 Due date Feb 16 In this chapter we learn how to compute the derivative of some functions using the de nition of the derivative One reason for doing this is to convince you that the rules and formulas for derivatives are not magical They have a solid foundation and can be explained with just a little bit of effort Learning should not just be a matter of memorizing mysterious formulas but it should rather be a matter of understanding them We start by recalling the following facts that we encountered in Chapter 2 gt Basic facts about derivatives The instantaneous rate of change of a function f with respect to x at a general point z is called the derivative of f at z and is denoted with f z m m fz 112 M For a given value 0 the derivative of f at 0 namely quotz07 gives the slope of the tangent line to the graph of f at the point mo7 fz0 Thus7 the equation of the tangent line to such a point is given by the formula 24 900 fQEOWE 900 Next7 we use the de nition of the derivative to learn how to differentiate functions of the following types 1 1 1 fa2 fa 1W a f9690a3 fW 1 where 04 is an arbitrary real number For each type of function7 the calculation of the limit has to be treated with a different technique Example 1 Let x m 4 0 Find constants A7 B7 and C such that Am 1 Bh C M h 7 W h 0 Show that the derivative of f is given by the expression f z 2x 8 2m 4 0 Find f 5 Write the equation of the tangent line to the graph of f at z 5 in the form y mm b Expanding the binomial z42 in the expression for the function f of Example 1 yields that x z28z 16 Hence the result f m 2x 8 also follows from the calculation carried out in Chapter 2 Example 15 1 Example 2 Let x 13 h 7 71 0 Find constants A B C and D such that x I m 2 14 B th Dh 0 Show that the derivative of f is given by the expression f z W 7 m 3 Find f 5 mm 0 Find constants A B C and D such that m h 7 m A t h Bz Ch D m 1 1 77 7242 2xz72 23 0 Show that the derivative of f is given by the expression f z Find ms and f 11 memorize the following formulas We can verify the 1 AB2A22ABB2 2 A7B2A272ABB2 Special product formulas The powers of certain binomials occur so frequently that we should m by performing the multiplications If A and B are any real numbers or algebraic expressions then 3 A B3 A3 3A2B 3le2 B3 4 A 7 B3 A3 7 3A2B 3AB2 7 B3 Visualizing a formula Many of the special product formulas can be seen as geometrical facts about length area and volume The ancient Greeks always interpreted algebraic formulas in terms of geometric gures For example the gure below AB2A22ABB2 shows how the formula for the square of a binomial formula 1 can be interpreted as a fact about areas of squares and rectangles Example 4 Let gz 0 Find constants A B C D E and F such that x 7 43 Pascal7s triangle The coef cients without sign of the expansion of a binomial of the form a i b can be read off the 71 th row of the following triangle named Pascal7s triangle after Blaise Pascal a 17th century French mathematician and philosopher To build the triangle start with 1 at the top then continue placing numbers below it in a triangular way Each number is simply obtained by adding the two numbers directly above it 710 711 712 713 714 715 WAmzBzCthEhFh2 0 Show that the derivative of g is given by the expression g m 3z 7 42 3x2 7 24m 48 0 Find g 6 and g 71 Let W Z1 aw071 23h 0 Show that h w 3gt2 w 3 I02 72 0 Show that the derivative of f is given by the expression f z W 72 m 3 3 Z13 Find H2 rlt gt L x p m 7 3 0 Show that x I127 x 71 xm73xmh73 xz73xh73gt 71 1 0 Show that the derivative of f is given by the expression f z 77z 7 3 32 2 x 7 3 m 7 3 2 Find f 7 and f 12 For Examples 1 57 see pages 78 81 of the text for some details to the algebra 42 gt Approximating a derivative At the moment it is too dif cult to compute general formulas for the derivatives of some important functions such as power trigonometric exponential and logarithmic functions Yet we routinely see these functions as buttons on our calculator The exercises below encourage us to approximate the value of the derivative of these functions at some selected values Example 7 Let x m 75 Approximate f 2 h 701 7001 70001 700001 7000001 000001 00001 0001 001 01 f2hf2 h h 95 7 95 h Example 8 Let gz lnz Approximate g 2 h 701 7001 70001 700001 7000001 000001 00001 0001 001 01 92 h 92 h ln2 h 7 ln2 71 Example 9 Let x 3m Approximate f 1 h 701 7001 70001 700001 7000001 000001 00001 0001 001 01 Example 10 Let gz Approximate g 0 h 701 7001 70001 700001 7000001 000001 00001 0001 001 01 gt Tangent lines continuity and differentiability In the following problems we practice computing equations of tangent lines Also7 we investigate further the notions of continuity and dil ferentiability of a function at a point Please refer back to Chapter 3 for the corresponding de nitions Example 11 The graph of a function hz and the coordinates of a point mo7 hz on the graphs of h are given below Find h m0 by analyzing the graph 39 h 3 4 W3 In the following problems you can use the fact that the derivative of az2bzc is fm 2am I See the calculation carried out in Chapter 27 Example 15 Example 12 Consider the function x 3x2 7 6x 7 10 Write the equation of the tangent line to the graph of f at z 72 in the form y mm b 7 for appropriate constants m and 1 Example 13 Consider the function gz 73m2 7x 7 6 Write an equation for its tangent line at z 1 For which values of y1 and yg does this tangent line go through the points 71731 and 4342 Example 14 If the tangent line to the function fz a at z 16 has equation y mz 57 nd a and m Hint to nd f m use the method illustrated in Example 4 Example 15 Consider the functions given below In each case7 nd all values of z where the derivative of the function is not de ned that is the points where the function is not di erentiable Is the function continuous at those points 7x1 ifmgi gx if 73ltmlt2 7x26 ifoQ hm 2 7 7m 10l Hint rst draw the graph of the equation y 2 7 7x 10 and then draw the graph of the function h