Formulas for derivativesChapter 5: Practice/review problemsThe collection of problems listed below contains questions taken from previous MA123 exams.Derivatives[1]. If f(x) = 6x2+ 3x − 1, find f′(x).(a) 6x + 1 (b) 12x + 3 (c) 12x −1 (d) 2x + 3 (e) 2x + 5[2]. If f(x) = x3+ 4x2+ 2x + 1 then f′(x) =(a) 3x2+ 8x + 3 (b) x2+ x + 1(c) 3x2+ 8x + 2(d) 3x2+ 8x + 1 (e) 3x2+ 4x + 1[3]. If f(x) = x3thenlimh→0f(1 + h) −f(1)h(Hint: Relate the limit to the derivative of f (x).)(a) 0 (b) 1 (c) 2 (d) 3 (e) 4[4]. Suppose f(t) = t3− t2+ t + 1. Find the limitlimh→0f(1 + h) −f(1)h(Hint: Relate the limit to the derivative.)(a) −1 (b) 0 (c) 1 (d) 2 (e) The limit does not exist[5]. If Q(s) = s7+ 1, findlimh→0Q(1 + h) − Q(1)h(a) 2 (b) 5 (c) 6 (d) 7 (e) 8[6]. If f(x) = |x − 1| find limh→0f(1 + h) −f(1)h(a) 3 (b) −3 (c) 1 (d) −1 (e) Does not exist[7]. Let f(x) = x |x| − x. Find the derivative, f′(0) , by evaluating the limitlimh→0f(h) −f(0)h(a) −2 (b) −1 (c) 0 (d) 1 (e) Does not exist61[8]. Let [[x]] denote the greatest integer function. Recall the definition:[[x]] equals the greatest integer less than or equal to x .How many points are there in the interval (1/2, 9/2) where the derivative of [[x]] is not defined?(a) 1 (b) 2 (c) 3 (d) 4 (e) 5The product rule[9]. Suppose that h(x) = f (x)g(x). Assume that f(2) = 3, f′(2) = −2, g(2) = 1, and g′(2) = 5. Find h′(2).(a) −20 (b) −17 (c) 11 (d) 13 (e) Cannot be determined[10]. If h(t) = (t − 1)(t + 1)(t2+ 1) then h′(2) equals(a) 0 (b) 4 (c) 8 (d) 16 (e) 32[11]. Let k(x) = (x + 3)(x + 4)(x + 1). Find k′(x).(a) 12 (b) 3x2+ 16x + 19(c) 3x2+ 18x + 20(d) 3x2+ 14x + 16(e) 1[12]. If R(x) = (x −2)(x2− 2)(x3− 2), find R′(2)(a) 0 (b) 12 (c) 48 (d) −8 (e) −6The quotient rule[13]. If f(x) =x − 1x + 1then f′(x) =(a)2x2+ 1(b)2(x + 1)2(c)−2(x + 1)2(d)−2x2+ 1(e)−2(x − 1)2[14]. Suppose that f(x) =x2+ 1x + 4. Find f′(−3).(a) −8 (b) −9 (c) −10 (d) −14 (e) −16[15]. Find Y′(s) if Y (s) =14s2−5s.(a)52s−3+ s−2(b) −12s−3+ 5s−2(c) −25s−3+ s−2(d)12s−3+ 5s−2(e) −2s−3− 3s−262[16]. If F (t) =3t + 1t − 1then F′(t) =(a) −4/(t − 1)2(b) −4/(3t + 1)2(c) −2/(t − 1)2(d) −3/(t − 1)2(e) −2/(t − 1)[17]. Let T (x) =g(x)f(x). If f(2) = 3, f′(2) = 4, g(2) = 5, and g′(2) = 6, find T′(2).(a)389(b)3825(c)225(d) −29(e) 38[18]. Evaluate the derivative, H′(1) ifH(s) =2ss + 1(a) 2/9 (b) 4/9 (c) 1/2 (d) 3/2 (e) 8/9[19]. Suppose the cost, C(q), of stocking a quantity q of a product equalsC(q) = 12 + 3q +8q.The r ate of change of the cost with respect to q is called the marginal cost. What is the marginal costwhen the cost equals 23 and th e cost is decreasing?(a) −5 (b) −1 (c) 0 (d) 1 (e) 5[20]. Suppose the cost, C(q), of stocking a quantity q of a product equalsC(q) =100q+ qFor which positive value of q is the tangent line to the graph of C(q) a h orizontal line?(a) 1/100 (b) 1/10 (c) 1 (d) 10 (e) 100[21]. Suppose u(t) and w(t) are differentiable for all t and the following values of the functions and derivativesare known: u(7) = 2, u′(7) = −1, w(7) = 1, and w′(7) = 9. Find the value of h′(7) whenh(t) =w(t) + 5u(t).(a) 3 (b) 6 (c) −3 (d) 12 (e) −6[22]. Suppose f(t) =F (t)tand F (1) = 2, F′(1) = 6. Find f′(1).(a) 2 (b) 4 (c) 1 (d) −4 (e) −163[23]. If f(x) =−xx2− 1then f′(x) =(a)−x2− 1(x2− 1)2(b)12x(c)−x2− 1x2− 1(d)x2+ 1x2− 1(e)x2+ 1(x2−1)2Tangent lines[24]. Find the equation of the tangent line to the graph of y = 2x3− 3x2+ 4x + 2 at x = 1.(a) y = x + 1 (b) y = 5x −4 (c) y = 5x − 3 (d) y = 4x − 2 (e) y = 4x + 1[25]. Which horizontal line is tangent to the graph of y = x3− x2− x + 2?(a) y = 0 (b) y = 1 (c) y = 2 (d) y = 3 (e) y = 5[26]. If g(t) =1t2+ 1, then the s lope of the tangent line to the graph of g(t) at t = 3 is(a) −125(b) −225(c) −150(d) −350(e) −425[27]. The equation of the tangent line to the graph of y = g (x) at x = 3 is y = 2 + 4(x − 3).What is the value of g′(3)?(a) −6 (b) 4 (c) −12 (d) 0 (e) 2[28]. If the line y = 3 + 4(x − 2) is tangent to the graph of g(x) at x = 2 and g(x) is differentiable at x = 2,then g(2) + g′(2) =(a) 2 (b) 3 (c) 4 (d) 6 (e) 7[29]. If the line y = 9 + 3(x − 4) is tangent to the graph of G(x) at x = 4 and G(x) is differentiable at x = 4,then G(4) − G′(4) equ als(a) 3 (b) 4 (c) 5 (d) 6 (e) 9[30]. The line y = −1 + 4(x −2) is tangent to the graph of g(x) at x = 2. If g(x) is differentiable at x = 2, andh(x) = xg(x), then h′(2) equals(a) 2 (b) 3 (c) 4 (d) 6 (e) 7[31]. LetH(s) =3(s − 1)2if s ≤ 15(s − 1)2if s > 1Find the equation of the tangent line to the graph of H(s) at s = 2 in th e (s, t) plane.(a) t = 3 + 6s (b) t = 3 −6s (c) t = 5 + 10(s − 2)(d) t = 5 + 10(s − 1) (e) The tangent line does not exist64[32]. LetH(s) = |s −1|Find the equation of the tangent line to the graph of H(s) at s = 0 in th e (s, t) plane.(a) t = 1 + s (b) t = 1 −s (c) t = 1 (d) t = s (e) The tangent line does not existThe chain rule[33]. If f(s) = (s2+ 5s + 4)3, find f′(s).(a) 3(s2+ 5s + 4)2(b) 3(s2+ s + 4)2(c) 2(s2+ 5s + 4) · (2s + 5)(d) 3(s2+ s + 4)2· (2s + 1)(e) 3(s2+ 5s + 4)2· (2s + 5)[34]. Find f′(1) where f(x) =√x4+ 3x2+ 5.(a) 1/3 (b) 2/3 (c) 1 (d) 4/3 (e) 5/3[35]. Suppose that h(x) = f (g(x)). Assume that f(3) = 6, f′(3) = 6, g(2) = 3, and g′(2) = 4. Find h′(2).(a) −30 (b) 24 (c) 18 (d) −20 (e) −15[36]. Suppose that f(x) = (x2− 5)3/2. Find f′(3).(a) 9 (b) 18 (c) 27 (d) 12 (e) 36[37]. Suppose that g(x) = [f(x)]3and the equation of the tangent line to the graph of f (x) at x = 2 isy = −1 + 4(x − 2). Find g′(2).(a) 15 (b) −15 (c) −1 (d) −12 (e) 12[38]. Suppose that f(t) = 12√t + …
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