You should be able to visually approximate the slope of a graph. The slope m of the graph of f at the point x, f x is given by
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1 Section. Te Tangent Line Problem r 5 sin, e, 88. r sin sin Parabola 9 9 Hperbola e ,,,, ortogonal 9. 5, 5,, 5, 5. Not multiples of eac oter; neiter parallel nor ortogonal 9.,,, 9, 8 parallel 9.,,,,. Not multiples of eac oter; neiter parallel nor ortogonal Section. Te Tangent Line Problem You sould be able to visuall approimate te slope of a grap. Te slope m of te grap of f at te point, f is given b m f f provided tis limit eists. You sould be able to use te limit definition to find te slope of a grap. Te derivative of f at is given b f f f provided tis limit eists. Notice tat tis is te same limit as tat for te tangent line slope. You sould be able to use te limit definition to find te derivative of a function. Vocabular Cec. calculus. tangent line. secant line. difference quotient 5. derivative. Slope is at,.. Slope is at,.. Slope is at,.. Slope is at,. 5. m sec g g m
2 9 Capter Limits and an Introduction to Calculus. m sec f f, m m sec g g m m sec m m sec m m m sec m sec g g g g 5, 9 9 m , ,.. m sec m 9 f m sec f f m , (a) At,, m. (b) At,, m.,
3 Section. Te Tangent Line Problem f m sec, m (a) At,, m. (b) At, 8, m. f m sec m f f f f,,,,, (a) At m (b) At m... f m sec m (a) At f f,, m. (b) At,, m. 7. f m sec m f f, (a) At m 5,, (b) At,, m 5..
4 9 Capter Limits and an Introduction to Calculus 8. f m sec f f m (a) At 5,, m (b) At 8,, m.. 9. f. f. f 7 5 (, ) 8 (, ) (, ) 5 Slope at, is. Slope at,. Slope at, is.. f. f (, ) 5 Slope at,. (, ) Slope at, is. f 8 Slope at,. (, ) f. f f f f f 5 5 g g 9 7. g lim 9
5 Section. Te Tangent Line Problem f f f f f f f f f. f f f.. f f f f f f
6 9 Capter Limits and an Introduction to Calculus. s s s s s s s s s s s s s s s s s s s s s s s s s s 5. f,, (a) (b) m sec f f m Tangent line: 5, (c) 5 (, ). f,, (a) (b) m sec f f m Tangent line:, (c) 8 (, ) 7. f,, (a) m sec f f m (c) (b) Tangent line: (, ) 5
7 Section. Te Tangent Line Problem f,, (a) (b) m sec f f m Tangent line: (c) 5 (, ) 5 9. f f f f Te appear to be te same f f f f Te appear to be te same.. f f f f Te appear to be te same.
8 9 Capter Limits and an Introduction to Calculus f f f f 8 Te appear to be te same.. Given line: m m tan since te lines are parallel. f f f m tan m tan Point:, f, Tangent line:. Since te tangent line is parallel to, te tangent line as a slope of f f f f f Te tangent line as slope m and passes troug te point,., tangent line at, m.
9 Section. Te Tangent Line Problem Given line: m m tan since te lines are parallel. f f f m tan m tan ± Points:, f, and, f, Tangent lines: and 8 8. Since te tangent line is parallel to, te tangent line as a slope of m. f f f Te tangent line as slope m and passes troug te point,. 8, tangent line at,
10 98 Capter Limits and an Introduction to Calculus 7. f f f f f as a orizontal tangent at,. 8. f f f f Impossible; no orizontal tangents 9. f f f f 9 9 ± f as orizontal tangents at, and,. 5. f f f 8 f, f as orizontal tangents at, and,.
11 Section. Te Tangent Line Problem (a) Quadratic model: Year Revenue, (b) Wen te slope is approimatel 99.. Tis represents a $99. million rate of cange of revenue in. (c) Tangent line: Wen, 558. (Model value) Te slopes are te same. 5. (a) N.p 8.5p. (c) 5 5 (b) Slope 5. for p 5 Slope 9. for p (d) Te rate of decrease in sales decreases as te price increases. 5. f 5 Using te definition of slope, ou obtain f 5. For, f > eigt increasing. For, f < eigt decreasing. 5. P True. Te slope is, wic is different for all. Using te definition of derivative, P. For, P > (profit increasing). For, P < (profit decreasing). 5. False. For eample, te tangent line to at, intersects te curve at, Matces (b). (Derivative is alwas positive, but decreasing.)
12 Capter Limits and an Introduction to Calculus 58. Matces (a). (Derivative approaces wen approaces.) 59. Matces (d). (Derivative is for <, for >.). Matces (c). (Derivative decreases until origin, ten increases.). Answers will var.. Answers not unique. Answers will var.. f and g (a) f (b) g (c) Answers will var. If n, n n. f 5 f g 5 g 5. f. f Vertical asmptotes:, Vertical asmptotes:, Horizontal asmptote: Intercept:, Horizontal asmptote: Intercepts:,,, 5 7. f Line wit ole at, Intercepts:,,,,
13 Section. Limits at Infinit and Limits of Sequences 8. f Line wit ole at, 8, i j,,,,,, i j,,,,,, i 7. u v 7 i 7. u v 8 j 7 8 j,,,, Answers will var. Section. Limits at Infinit and Limits of Sequences Te limit at infinit lim f L means tat f gets arbitraril close to L as increases witout bound. Similarl, te limit at infinit lim f L means tat F gets arbitraril close to L as decreases witout bound. You sould be able to calculate limits at infinit, especiall tose arising from rational functions. Limits of functions can be used to evaluate limits of sequences. If f is a function suc tat lim f L and if a n is a sequence suc tat f n a n, ten lim a n n L. Vocabular Cec. limit, infinit. converge. diverge. Intercept:, Horizontal asmptote: Matces (c).. Horizontal asmptote: Matces (a).. Horizontal asmptote: Vertical asmptote: Matces (d).
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