rfs - MA1021Calculus I 1 Find dy/dx if(a(b(c(d(e(f(g y = x3...

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MA1021—Calculus I Final Review Solutions 1. Find dy/dx if (a) y = x 3 + 10 x 2 - 50 x + 100 = dy dx = 3 x 2 + 20 x - 50. (b) y = ( x 2 + 25)( x 3 - x ) = dy dx = 2 x ( x 3 - x ) + ( x 2 + 25)(3 x 2 - 1). (c) y = 1 + sin(3 x ) 1 - sin(5 x ) = dy dx = (1 - sin(5 x ))(3cos(3 x )) - ( - 5cos(5 x ))(1 + sin(3 x )) (1 - sin(5 x )) 2 . (d) y = p x 3 + 25 = dy dx = 3 x 2 2 x 3 + 25 . (e) 3 x 2 + xy = y 2 + 16 = dy dx = 6 x + y 2 y - x . (f) x = sin( y 2 ) = dy dx = 1 2 y cos( y 2 ) . (g) y = 2 x - 3 = dy dx = 1 2 x - 3 . 2. Compute the following limits. (a) lim x 2 x 3 - 3 x 2 + 2 x x - 2 = lim x 2 x ( x - 2)( x - 1) x - 2 = lim x 2 x ( x - 1) = 2 . (b) lim θ 0 1 - cos( θ ) θ 2 = lim θ 0 1 - cos 2 ( θ ) θ 2 (1 + cos( θ )) = lim θ 0 sin 2 ( θ ) θ 2 1 (1 + cos( θ )) = (1) 2 1 2 = 1 2 ; (c) lim x 4 - | x - 4 | x - 4 = lim x 4 - - ( x - 4) x - 4 = lim x 4 - - 1 = - 1. 3. Use the definition to compute the derivative of f ( x ) = 2 x - 3 . f 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h = lim h 0 p 2( x + h ) - 3 - 2 x - 3 h = lim h 0 (2( x + h ) - 3) - (2 x - 3) h p 2( x + h ) - 3 + 2 x - 3 · = lim h 0 2 h h p 2( x + h ) - 3 + 2 x - 3 · = lim h 0 2 p 2( x + h ) - 3 + 2 x - 3 · = 2 ( 2 x - 3 + 2 x - 3) = 1 2 x - 3
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4. Consider the function f ( x ) = x 3 + 3 x 2 - 9 x - 10 = f 0 ( x ) = 3 x 2 + 6 x - 9 . (a) Find the equation of the tangent line to the graph y = f ( x ) at x = 2. You have the slope: f 0 (2) = 15 and a point: (2 ,f (2)) = (2 , - 8), so the equation for the tangent line is ( y + 8) = 15( x - 2) or y = 15 x - 38. (b) Determine intervals on which the function is increasing as well as those on which it is decreasing. f 0 ( x ) = 3( x + 3)( x - 1) is positive for x < - 3 and x > 1 so f ( x ) is increasing here. f
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This note was uploaded on 09/02/2010 for the course MATH SMUD 206 taught by Professor Condon during the Spring '10 term at UMass (Amherst).

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rfs - MA1021Calculus I 1 Find dy/dx if(a(b(c(d(e(f(g y = x3...

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