Business Calculus MATH 126
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This 13 page Class Notes was uploaded by Breana Ullrich on Monday October 19, 2015. The Class Notes belongs to MATH 126 at Radford University taught by William Case in Fall. Since its upload, it has received 48 views. For similar materials see /class/224679/math-126-radford-university in Mathematics (M) at Radford University.
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Date Created: 10/19/15
Section 91 Limits Example of limits Find the limit x2 1 Find the limit of the function as x approaches 1 Ix 9 99 999 11 101 1001 fx 1810 19801998 2210 2020 2002 The values in the table seem to approach 2 as X approaches 1 There the limit of function as X approaches 1 is 2 This is written symbolically as lirrlwc2 1 2 De nition 1 lim f x f c Thus the limit in the above example can be found by using definition 1 lirr11fxf112 1112 Example 1 Find lirr31x3 Solution lirr31x3 33 27 Example 2 Find lirn 3x2 8x Solution 1113 3x2 8x 3 42 8 4 306 32 48 32 16 Example 3 Find liIni x91 x 1 Solution liIni i 2 Unde ned H1 x 1 1 1 0 Limit does not exist Example 4 Find linsh x 7 Solution linshx 7 5 7 x 2 Unde ned Limit does not exist Example 5 Find 11m 3 3 x93x 9 Solution lirnxzi3 x93 x 16 3 1 1 llm 11m7 7 9 xx3x 3x3 xx3x3 33 6 In example 5 if you substitute 3 in for x before you reduce the polynomial down you will get an undetermined form Example 6 2 x Find liIn x92 x2 x2 1 1 1 Solution 11m lim 1nn 7 Hzx 4 H3x 2x2 st2 22 4 Example 7 2 Find unfit2 r 1 tgt1 2 Solution limw1j W 123 H1 t2 1 H1 t1t1 H3 t1 11 2 Example 8 Find 1nn 4 5 x971 3 Solution liml43x5L545 9 x 31 4 Z Evaluating a limit from a graph Example 9 a V Find lxiLIZIfOC Solution lXLIIZIfOC 2 b Find lilgfoo Solution fx 4 c Find f x Solution Iii f x 1 Left hand limit The y value X approaches from the left d Find fx Solution lim f x 4 Right hand limit The y value X approaches from the right H0 6 Find lim f x Does Not Exist The limit doesn t exist since the right hand x90 and the left hand limits are different Section 93 Derivatives The derivative is the slope tangent line to the curve Limit Definition of a Derivative yz yl 2 fxh fx 2 fxh fx xz x1 xh x h f39x slope m Example 1 Given f x 3x 2 nd f x using the limit definition of a derivative f39x lim hgt0 fxh fx m3xh 2 3x 2 m3x3h 2 3x2 h JkJgt h k o h lim lim3 3 kw h kao Example 2 Given f x 5x 3 find f 39x using the limit definition of a derivative im5xh3 5x3 k lm5x5h3 5x 3 hi h H h 1 1 l l im fxh fx f Jo1 h limit lim5 5 0 hgt0 h hgt0 0 o Example 3 Given f x x2 2 find f 39x using the limit definition of a derivative f x 13301 fxhfx h 2 2 thxh 2 x 2 kw h 2 thxhxh 2 6 2 kw h x2xhxhh2 2 x22 lim h h l x22xhh2 2 x22 lim kw h i 2xhh2 llm7 kw h limwlim2xh 2x0 2x hgt0 h hgt0 Example 4 Given f x x2 5 nd f x using the limit definition of a derivative f x 13301 xh2 5 x2 5 h fxhfx h lim kw 2 hmxhxh5 x 5 kw h i x2xhxhh25 x2 5 11In kw h i x22xhh25 x2 5 11In kw h i 2xhh2 limi kw h thh2xh kw h lim2xh kw 2x0 2x Power Rule Function Derivative fx3x2 f x3 fxx2 2 f x2x fx x3 3x2 f x 3x2 6x fx x4 fx 4963 When looking at the changes between the function and the derivative it can be observed that the power decreases by a power one and the leading coefficient is multiplied by the exponent Using this relationship the power rule for derivative can be developed General Power Rule Given f x ax f 39x nax H Example 5 Given fx2x5 nd f x f x 5 2x 10x4 Example 6 Given fx 3x4 nd f OC f x 4 3x 12x3 Example 7 Given fx x3 2x2 3x nd f m f x3x37122x3x1 123x2 4x3 Section 101 A Increasing and Decreasing Functions Example 1 Give the intervals where the function is increasing and decreasing tmsxa x Decreasing oo Increasin 39 2 oo 2 2 Example 2 Give the intervals where the function increasing and decreasing rm xle a Increasing 00 1U1 Decreasing 11 Test for increasing and decreasing functions Let f be a differentiable function on the interval 1 If f x gt Ofor all X in ab then f is increasing 0n ab 2 If f x lt Ofor all X in ab then f is decreasing 0n ab 3 If f x Ofor all X in ab then f is constant on ab Example 3 Give the intervals where the function is increasing and decreasing f x x2 4x f x2x 4 2x 40 2x 4404 2x4 2x4 7 3 x2 f 1 2 1 4 2 4 6 f 323 42 Example 4 Find the intervals where the function is decreasing and increasing fx x3 4 f39x 3x2 f 1 3 1gt2 3 f 1 312 3 Example 5 Find the intervals where the function is increasing or decreasing f x x3 3x2 f39x3x2 6x 3x2 6x0 3xx 20 3x0 or x 20 3ic90rx 2202 3 3 x0 x2 f 1 3 12 6 1 36 9 f391 312 61 3 6 3 f 3 332 63 27 18 9
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