Rates of change and derivativesChapter 2: Practice/review problemsThe collection of problems listed below contains questions taken from previous MA123 exams.Average rates of change (Word Problems)[1]. A train travels from A to B to C. The distance from A to B is 10 miles and the distance from B to C is40 miles. The average velocity from A to B was 20 miles per hour and the average velocity f rom B to Cwas 40 miles per hour. What was the average velocity from A to C in miles per hour?(a) 180/5 (b) 90/3 (c) 100/3 (d) 180/3 (e) 100/5[2]. A train travels from city A to city B. It leaves city A at 10:30 am and arrives at city B at 1:30 pm. Thedistance between the cities is 150 miles. What was the average velocity of the train in miles per hour?(a) 60 (b) 150 (c) 50 (d) 75 (e) 130[3]. A train travels from city A to city B to city C. The distance from A to B is 20 miles. The distance fromB to C is 45 miles. The train took 1 hour for the trip from A to B, stopped at city B for 30 minutes,and then went from B to C at an average velocity of 30 m iles per hour. What was the average velocityof the train for the entire trip (in miles per hour)?(a) 65 (b) 25(c)652(d) 50(e)653[4]. A train travels from A to B to C. The distance from A to B is 30 miles and the distance from B to C is80 miles. The train leaves A at 10:00 AM and arrives at C at 3:00 PM. The average speed from A to Bwas 30 miles per hour. What was the average speed from B to C in m iles per hour?(a) 20 (b) 25 (c) 30 (d) 35 (e) 40[5]. A train travels from city A to city B. The cities are 600 miles apart. The distance from city A at t hoursafter the train leaves A is given by d(t) = 50t + t2.What is the average velocity of the train in miles per hour during the trip from A to B?(Hint: First find how long it takes for the train to get from A to B.)(a) 50 (b) 55 (c) 60 (d) 65 (e) 70[6]. John leaves at 9:00 am and drives from Lexington to Ashland arriving at 11:00 am. He stops for twohours since his girlfriend Mary is not yet ready. Then they drive together from Ashland to Columbusarriving at Columbus after a three-hour d rive. The distance from Lexington to Ashland is 110 miles andthe distance from Ashland to Columbus is 130 miles. Find the average velocity of John’s car in miles perhour for the entire trip (including the two hour stop) correct to two decimal places.(a) 33.81 (b) 33.42 (c) 35.00 (d) 34.29 (e) 34.4720Average rates of change[7]. If g(x) = (x −1)2what is the average rate of change of g(x) with respect to x as x changes from − 3 to 3?(a) −4 (b) −2 (c) 0 (d) 2 (e) 4[8]. Suppose that h(t) =2t. Find th e average rate of change of h(t) from t = 5 to t = 10.(a) −.05 (b) −.04 (c) .05 (d) .04 (e) .02[9]. Find the average rate of change of the function R(t) =√2t + 7 as t changes from 1 to 9.(a)13(b)12(c)14(d) 4 (e) 2[10]. If g(x) = |x −7| what is the average r ate of change of g (x) with respect to x as x changes from −3 to 3?(a) −2 (b) −1 (c) 0 (d) 1 (e) 2[11]. Find the average rate of change of the function G(t) =t2− 1as t changes from −1 to 2.(a) 0 (b) 1 (c) 2 (d) 3 (e) 4[12]. Let g(s) = s2−3s + 1. Find a value A ≥ 0 such that the average rate of change of g(s) from 0 to A equals8.(a) 0 (b) 8 (c) 11 (d) 15 (e) 22[13]. Suppose f(t) = t3+ 1. Find a value A greater than 0 such that the average rate of change of f(t) from 0to A equals 2.(a) 1 (b)√2(c)√3(d) 2(e)√5Difference quotients[14]. Computef(2 + h) − f(2)hwhere f(x) = 3x2+ 1.(a) 12 (b) 12 + h (c) 12 + 2h (d) 12 + 3h (e) None of the above[15]. What is the average rate of change of g(s) = s2− 4 as s changes from 1 to 1 + h?(a) 6 + 3h (b) 2 + h (c) 4 + 2h (d) 2 (e) h[16]. Let f(x) = 2x2− 3x. Find the average rate of change of f (x) from x = 3 to x = 3 + h.(a) 9 − h (b) 9 + h (c) 9 (d) 9 − 2h (e) 9 + 2h21[17]. Let g(t) = (t − 5)2+ 1. What is the average rate of change of g(t) as t changes from 4 to 4 + h?(a) h2− 2h(b) h + 2(c) h2+ 2h(d) h − 2 (e) 1[18]. If f(t) = 3t2+ 4 thenf(1 + h) − f(1)h=(a) 4 + 3h (b) 3 + 4h (c) 6 + 3h (d) 8 + 3h (e) 8 + 4h[19]. If f(t) = 1/t th enf(t + h) − f(t)h=(a) 1/(h2)(b) 1/(t(t + h)) (c) −1/(t(t + h))(d) 1/(t(t − h)) (e) −1/(t(t − h))Instantaneous rates of change[20]. Consider a triangle with base x and height 2x. Find the instantaneous rate of change of the area of thetriangle with respect to x when x = 5.(a) 1 (b) 2 (c) 5 (d) 10 (e) 20[21]. Find the instantaneous rate of change of the fun ction H(t) = t3at t = 2.(a) 2 (b) 3 (c) 8 (d) 12 (e) 27In what follows, you may use the following formula for the derivative of a quadratic function.If p(x) = Ax2+ Bx + C, then p′(x) = 2Ax + B.[22]. If g(s) = 3s2+ s −2 what is the value of g(s) when the instantaneous rate of change of g(s) with respectto s equals 1?(a) − 2 (b) −1 (c) 0 (d) 1 (e) 2[23]. If g(s) = 3s2+2s−2 what is the value of s for which the instantaneous rate of change of g(s) w ith respectto s equals 8?(a) −2 (b) −1 (c) 0 (d) 1 (e) 222[24]. Suppose the pr ice of a good is given by th e quadratic function P (t) = 2.58 + .14t + .01t2. What is theinstantaneous r ate of change in the price when t = 3?(a) .18 (b) .20 (c) .22 (d) .24 (e) .26[25]. Let g(x) = x2+ 4x + 5. Find a value of c between 1 and 10 such that the average rate of change of g(x)from x = 1 to x = 10 is equal to the instantaneous rate of g(x) at x = c.(a) 4.75 (b) 5.0 (c) 5.25 (d) 5.5 (e) 5.75[26]. Find a n onnegative number A such that the average r ate of change of F (t) = t2− 2t + 1 from t = 1 tot = A equals the instantaneous rate of change of F (t) at t = 2.(a) A = 0 (b) A = 2 …