Computing some derivativesChapter 4: Practice/review problemsThe collection of problems listed below contains questions taken from previous MA123 exams.Computing some derivative s[1]. If f(x) = (x + 3)2thenf(x + h) − f(x)h=(a) 2x + h (b) 2x + 3 + h (c) 2(x + 3) + h(d) 2(x + 3) (e) 2x + 8 + h[2]. If f(x) = (x + 6)2, findf(x + h) − f(x)h(a) 2x + 2h + 12 (b) 2x + h − 2 (c) 2x + 2h + 2(d) 2x + h + 12 (e) 2x + h − 12[3]. If F (t) =3t + 1then the slope of the tangent line to the graph of F (t) at t = 2 is(a) −1/3 (b) −1/2 (c) 0 (d) 1/3 (e) 1/2[4]. Suppose that f(x) =2x + 3. Findf(x + h) − f(x)h.(a)−2(x + 3)2(b)−2h(x + 3)2(c)2(x + h + 3)(x + 3)(d)−2(x + h + 3)(x + 3)(e)2(x + 3)2[5]. Evaluate the limitlimh→0f(3 + h) − f(3)hwheref(x) =√x + 1(a) 1/6 (b) 1/5 (c) 1/4 (d) 1/3 (e) 1/2[6]. If F (s) =√2s + 2, find F′(1).(a)12(b)12√2(c)1√2(d)32√2(e)3247[7]. The equation of the tangent line to the graph of w =√t + 1 at t = 3 is(a) w = 2 + (1/3)(t − 3) (b) w = 2 + (1/4)(t − 3) (c) w = 3 + (1/4)(t −3)(d) w = 3 + (1/6)(t − 8) (e) w = 3 + (1/3)(t − 8)Approximating some derivatives (optional)[8]. Suppose f (x) = 2x. Use the definition of the derivative and a calculator to find the approximate value ofthe derivative of f at x = .4. Select the answer that best approximates the derivative.(a) .43 (b) .53 (c) .63 (d) .93 (e) 1.13[9]. Suppose f(x) = log(x) where log(x) denotes th e base 10 logarithm. Use the definition of the derivativeand a calculator to find the approximate value of the derivative of f at x = 2. Select the answer thatbest approximates the derivative.(a) .102 (b) .145 (c) .180 (d) .217 (e) .378[10]. Let f(x) = 2x.Use a calculator and the definition of the derivative as a limit to estimate the value of f′(1).(a) 1.386 (b) 2.296 (c) 4.768 (d) 5.545 (e) 8.047[11]. Let f (x) = ln(x + 2) + 1. Use the limit definition of the derivative and a calculator to estimate f′(4).Your answer should be correct to four decimal places.(a) 0.1667 (b) 0.2500 (c) 0.1429 (d) 0.2000 (e )
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