Differentiation Using Product and Quotient Rule 1

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Esmond Copeland
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1 Differentiation Using Prouct an Quotient Rule ( + 1)( + + 1) (7 + 15) ( ) ( + 7) (5 + 14) ( 1) (10 + ) ( + 1) ( 1) (5 + ) 14. 4( 1) ( + ) ( + 1) 15. ( ) ( + ) ( + 1) ( 1) ( + 4)

2 1. Fin the horizontal tangents of: y x 4 x + Horizontal tangents occur when slope zero. y 4x 4x 0 x(x + 1)(x 1) 0 x 0, 1, 1 Points: (0,) ( 1,1) (1,1) (The function is even, so we only get two horizontal tangents.). Fin equations of the tangent line an normal line to the curve y x 1+x at the point (1, ½). slope of the tangent line at (1, ½) : tangent line at (1, ½): y (x 1) y 1 4 x + 4 normal line at (1, ½): y 1 4(x 1) y 4x 7

3 . At what points on the hyperbola xy 1 is the tangent line parallel to the line x + y 0? Since xy 1 can be written as y 1/x, we have: y x 1(x 1 ) 1 x Let the x-coorinate of one of the points in question be a. Slope of the tangent line at that point is 1 a, an that has to be equal to the slope of line x + y 0 1 a or a ± the require points are: (, 6) an (-, -6)

4 Prouct & Quotient Rules with Trig 1. State the Prouct Rule: f x g x x [f(x)g(x)] f (x)g(x) + f(x)g (x) x. State the Quotient Rule: x f x g x x [f(x) g(x) ] f (x)g(x) f(x)g (x) g (x). Fin the erivative of each. (a) f x 6x 5 x (b) h t t sin t t cost f (x) 6(x ) + (6x + 5)(x ) h (x) sin t + t cos t + t cos t t sin t 4x + 15x 18 sin t + 4 t cos t t sin t (c) f x x cot x f (x) 4x cot x x csc x () f x x tan x sin x 1 f (x) (sin x + 1)(1 + sec x) (x + tan x)(cos x) (sin x + 1) sin x sec x sec x x cos x (sin x + 1) x x (e) f x x x 1 x 1 (x 1)(x + 1) (x x )(x) f x x (x) [ (x + 1) ] [x + x + 1] + [ x ] [x + 1] + 1 (f) f x tan xsin x f (x) sec x sin x + tan x cos x sec x tan x + sin x

5 cos x x (g) f x f (x) ( sin x)(x ) (cos x)(x) (x ) x sin x x cos x x 4 (h) h x csc x h (x) x [( 1 sin x ) ( 1 sin x )] 4. Evaluate f P ( π, 1) cos x ( sin ) ( 1 x sin x ) + ( 1 1 cos x ) (0 sin x sin ) x cot x sin x x. 4 if f x sin xsin x cos x, then fin the equation of the tangent line at 4 f (x) cos x(sin x + cos x) + sin x(cos x sin x) sin x cos x + cos x sin x ( ) ( ) + ( ) ( ) 1 tangent line at P ( π 4, 1) : y x + (1 π 4 ) 5. Fin the equation of the tangent line to f x x 1 x 1 at the point where f x crosses the x- axis. x intersection: (x 1)(x + 1) 0 P(1,0) f (x) (1)(x + 1) + (x 1)(x) f (1) tangent line at P(1,0): y x

6 6. Fin the equation of the tangent lines to the graph of all parallel lines have the same slope 1 y x x 1 that are parallel to the line y x 6. slope of the tanget line is 1 y (1)(x 1) (x + 1)(1) (x 1) (x 1) (x 1) 1 (x 1) 4 (x 1) ± x 1 & x tangent line at ( 1,0) or (,): y 1 x 1 & y 1 x If f x x x an g. f ()(x + ) (x)(1) (x) (x + ) 6 (x + ) an g x 5x 4 x, verify that f x g x, an explain the relationship between f g (5)(x + ) (5x + 4)(1) (x) (x + ) 6 (x + ) If f (x) g (x), then f(x)an g(x)only iffer by a constant. Proof: x (x + ) 6 6 f(x) x + x + x + + g(x) 5x + 4 x + 5(x + ) 6 x + 6 x + + 5

7 8. Determine whether there exist any values of x in the interval 0, such that the rate of change of f x sec x an the rate of change of g x csc x are equal. f (x) x [ 1 cos x ] (0)(cos x) (1)( sin x) sin x cos x cos x g (x) x [ 1 sin x ] sin x cos x cos x sin x (0)(sin x) (1)(cos x) sin x cos x sin x sin x cos x sin x cos x x π 4 & x 7π 4 9. Sketch the graph of a ifferentiable function f such that f 0, f 0 for x, an f 0 x for answers will vary 10. If g, g, h 1, an h 4, fin f for g x (a) f x g x h x (b) f x 4 h x (c) f x () f x g xh x h x f () 0 f () 4 f () 10 f () 8

8 11. Fin a secon-egree polynomial f x ax bx c such that f x has an x-intercept at x 1, an the graph of f x has a tangent line with a slope of 10 at the point,7 point of intercept: a + b + c 0 graph passes through the point (,7): 4a + b + c 7 tangent line at (,7) has the slope of 10: f (x) ax + b 4a + b 10 a + b + c 0 4a + b + c 7 4a + b 10 a b c 1 f(x) x x 1 1. If the normal line to the graph of a function f at the point 1, passes through the point 1,1, then what is the value of f 1? f (1) is the slope of tangent line at x 1. It is negative reciprocal of the normal line slope: m t 1 m n m n ( 1) f (1) m t 1. If y xsin x, fin y x y sin x x cos x x

9 14. Fin the following 999 (a) 999 cos x x (c) x sin x y cos x y sin x y cos x y sin x y (4) cos x y y (999) y sin x y sin x y cos x y sin x y cos x y (4) sin x y y (999) y cos x 15. If sin x sin x cos x an cos x cos x sin x ( (re)memorize these), use these ientities to evaluate the following using the prouct rule. (a) sin x (b) cos x x x [sin x] [ sin x cos x] x x (cos x)(cos x) + (sin x)( sin x) (cos x sin x) cos x b. [cos x] x x [cos x] x [cos x sin x] [(cos x)(cos x) (sin x)(sin x)] x [( sin x)(cos x) + (cos x)( sin x)] [(cos x)(sin x) + (sin x)(cos x)] sin x cos x sin x cos x sin x sin x sin x

10 SOLUTIONS (xx + h) xx 1. lim h 0 h xx xx tan (xx + h) tan h. lim 1 tan xx h 0 h ssssss xx (xx + h) 4 xx 4. lim h 0 h xx4 4xx (xx + h) + (xx + h) xx xx 4. lim h 0 h (xx + xx) 4xx + sin[(xx + h)] sin xx 5. lim sin xx cos xx h 0 h cos(xx + h) cos xx 6. lim h 0 h cos xx xx sin xx xx 7. lim 9 xx lim xx xx xx xx xx 6 xx xx 5 8. lim xx 5 xx 5 xx 10 xx5 xx 5 9. lim xx 5 xx 5 lim xx 5 xx 5 xx 5 xx 10 xx5 sin xx sin ππ 10. lim xx ππ/ xx ππ sin xx xxππ/ cos(π/) 1/ sin xx 11. lim xx ππ/ xx ππ sin xx cos(π/) 1/ xxππ/ 1. cos xx lim xx ππ/4 xx ππ cos xx sin π xxππ/ sin xx lim xx 0 xx lim sin xx sin 0 xx 0 xx 0 sin xx cos 0 1 xx0 sin xx 14. lim xx ππ xx ππ lim sin xx sin ππ xx ππ xx ππ sin xx cos ππ 1 xxππ

AP Calculus AB Unit 2 Assessment

Class: Date: 203-204 AP Calculus AB Unit 2 Assessment Multiple Choice Identify the choice that best completes the statement or answers the question. A calculator may NOT be used on this part of the exam.