University:
Winston-Salem Forsyth County Schools
Course:
AP Calculus AB
Academic year:

2014
Views:

299

Pages:

2
Author:

Makhi Browning

Insight Builders #2c ANSWERS Due: September 19, 2014 1 . Let f be the function given by f (x) = x 2 − 3x + 2 . (a) Find the domain of f. (b) Find the slope of the secant line joining ( 3, f (3)) and ( 4, f (4)) . (c) Let g(x) = f (x) + 2 . Find the slope of the secant line joining ( 3, g(3)) and ( 4, g(4)) . (d) Let h(x) = f (2x) . Find the slope of the secant line joining (1.5, h(1.5)) and ( 2, h(2)) . (a) x 2 − 3x + 2 ≥ 0 (x − 2)(x − 1) ≥ 0 3 x ≤ 1 or x ≥ 2 f (3) = 9 − 9 + 2 = 2 (b) f (4) = 16 − 12 + 2 = 6 m= Δy = Δx 2 6− 2 = 6 − 2 ≈ 1.035 4−3 (c) slope will be the same as in (b) or g(3) = 2 + 2 g(4) = 6 + 2 m= (d) Δy = Δx ( m= ) ( 6+2 − 4−3 2+2 )= 2 6 − 2 ≈ 1.035 h(2) − h(1.5) f (4) − f (3) = = 2 − 1.5 .5 6− 2 =2 .5 ( 6− 2 ) 2 2. Let h be a function defined for all x ≠ 0 such that h(4) = −3 and the derivative of h is given by x2 − 2 for all x ≠ 0 . h′(x) = x (a) Find all values of x for which the graph of h has a horizontal tangent, and determine whether h has a local maximum, a local minimum, or neither at each of these values. Justify your answers. (b) On what intervals, if any, is the graph of h concave up? Justify your answer. (c) Write an equation for the line tangent to the graph of h at x = 4 . (d) Does the line tangent to the graph of h at x = 4 lie above or below the graph of h for x > 4 . Why? (a) If there is a horizontal tangent then the slope must be zero. - + - + x2 − 2 =0 x 2 x −2=0 x=± 2 h′(x) = 0 The graph of h has a horizontal tangent at x = ± 2 . h has a local maximum at x = 0 , where h′ changes from positive to negative, although the derivative is undefined at x = 0 . h has local minima at x = ± 2 , where h′ changes from negative to positive. x2 − 2 2 = x− x x 2 h′′ = 1 + 2 x b) h′(x) = ( ) h′′ ( 2 ) = 2 h′′ − 2 = 2 + + 0 h′′ is undefined at x = 0 , an positive elsewhere, so h is concave up everywhere except at x = 0 . c) h′(4) = 16 − 2 7 = and (4,−3) is a point on h. 4 2 7 y + 3 = (x − 4) 2 d) Since h is concave up, the tangent line lies below the curve.