MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable CalculusFall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.01 REVIEW PROBLEMS AND SOLUTIONS Unit I: Differentiation R1-0 Evaluate the derivatives. Assume all letters represent constants, except for the independent and dependent variables occurring in the derivative. a) pV1 =nRT, dp= b)m = /dm =?dV 1 /c2 d R= dR =?cwosin(2k + 1)ac)R= + =?--RI-1 Differentiate: a) sin b)sin2(vr) c)Xz/Stan d) 2+2 e)c( 1) f)cos3(v ) g)tan( s3) h) sec2(3s+ 1) R1-2 Consider f () = 2z2+ 4z + 3. Where does the tangent line to the graph of f(z)at x = 3cross the y-axis? R1-3 Find the equation of the tangent to the curve 2s2+ zy-y +2x-3y =20 at the point (3,2). R1-4 Define the derivative off(x) .Directly from the definition, show that f'(z) = cos iff(x)=sin .(Youmay usewithout proof: lim sn = ,lim coh-1 0). h-+o h h-+O h R1-5 Find all real so such that f'(xo) = 0: a) f(s) = X- + 1 b)f()= +cosa R1-6 At what points isthe tangent to the curve y.+ XyX+ -3 = 0 horizontal? R1-7 State and prove the formula for (uv)' in terms of the derivatives of u and v. You may assume any theorems about limits that you need. R1-8 Derive a formula for (xl)' . R1-9 a) What is the rate of change of the area Aof asquare with respect to its side x ? b)What is the rate of change of the area Aofacircle with respect to its radius r? c) Explain why one answer is the perimeter of the figure but the other answer is not.1 REVIEW PROBLEMS AND SOLUTIONS RI-10 Find all values of the constants c and dfor which the function f( = +1, x cz+d,z <1 will be (a) continuous, . (b) differentiable. R1-11 Prove or give a counterexample : a) If f(z) is differentiable then f(z) is continuous. b) If f(z)is continuous then f(z)is differentiable. -sin;, z <r Rl-12 Find all values of the constants a and b so that the function f() = az+b, z >ir will be (a) continuous; (b) differentiable. R1-13 Evaluate li sin(4 z) (Hint: Let 4z = t.)2-+0 Z Unit 2: Applications of Differentiation R2-1 Sketch the graphs of the following functions, indicating maxima, minima, points of inflection, and concavity. a) f(z) = ( -1)2(z+2) b)f(z) =sin 2z, 0 <s < 2r c) f ()= Z + 1/z2 d) f(z) = z+sin 2z R2-2 A baseball diamond is a 90 ft. square. Aball is batted along the third base line at a conistant speed of 100 ft. per sec. How fast is its distance from first base changing when a) it is halfway to third base, b) it reaches third base ? R2-3 If x and y are the legs of a right triangle whose hypotenuse is v1 , find the largest value of 2z +y. R2-4 Evaluate the following limits: cos a sin : a) lim os b) lim s X-0 _4f+ 2- c) lim 017..-3+2 -oo 10t7 +6Z10 -3 -5zX R2-5 Prove or give a counterexample: a) If f'(c) = 0 then f has a minimum or a maximum at c. b) If fhas amaximum at cand if f isdiferentiable at c,then f'(c) = 0. R2-6 Let f(s) = 1-2 /3. Then f(-l) = f(1) = 0and yet f'(z) 0for 0 < z < 1.Find the maximum value of f(z) on the real line, nevertheless. Why did the standard method fail? R2-7 A can is made in the shape of a right circular cylinder. What should its proportions be, if its volume is to be 1and one wants to use the least amount of metal?REVIEW PROBLEMS AND SOLUTIONS R2-8 a) State the mean value theorem. 1 b) If f'(x) = 1+ z and f(1) = 1, use the mean value theorem to estimate f(2). (Write your answer in the form a < f(2) </.) R2-9 One of these statements is false and one is true. Prove the true one, and give a counterexample to the false one. (Both statements refer to all z in some interval a < x < b.) a) If f'(x) > 0, then f(x) is an increasing function. b) If f(z) is an increasing function , then f'(x) > 0. R2-10 Give examples (either by giving a formula or by a carefully drawn graph ) of a) A function with a relative minimum, but no absolute maximum on 0 < z. < 1. b) A function with a relative maximum but no absolute maximum on the interval 0<x<1. c) A function f(x) defined on 0 < x 1 , with f(0) < 0, f(1) > 0, yet with no root on 0 < <l. d) A function f (x) having arelative minimum at 0, but the following is false: f'(0) = 0. Unit 3: Integration R3-1 Evaluate: j sin dx, + dx, j 2--+ dz. R3-2 Egbert, an MIT nerd bicyclist, is going down a steep hill. At time t = 0, he starts from rest at the top of the hill; his acceleration while going down is 3t2 ft./sec2, and the hill is 64 ft. long. If the fastest he can go without losing control is 64 ft./sec., will he survive this harrowing experience? (A nerd bicycle has no brakes.) R3-3 Evaluate f Z Zsdz directly from the definition of the integral as the limit of a sum. You may use the fact that 6k=1 R3-4 If f is a continuous function, find f(2) if-a) f(t)dt = (1+ x) b) f(t)dt = x 2(1+ ) ) j . ' t x2(1+ ) R3-5 The area under the graph of f(x) and over the interval 0 5 z < a is -1 + a + sina+ 1cos a2 4 2 2 Find f (r/2).REVIEW PROBLEMS AND SOLUTIONS RS-6 Use the trapezoidal rule to estimate the sum -iV+ 2-... + I-.Is your estimate high or low? Explain your reasoning. R3-7 Find the total area of the region above the graph of y= -2x and below the graph of y = X -z 2. R3-8 Use the trapezoidal rule with 6 subintervals to estimate the area under the curve y= ,-3 5 z5 3.(You may use: /2s 1.41, v u 2.24, V/I f 3.16. Is your estimate too high or too low? Explain how yoU know.) R3-9 Fill in this outline of a proof that F'(s)ds = F(b) -F(a). Supply reasons. a)Put (s) = F'(t)dt. Then G'(z) =F'(s). b) Therefore G(s)= F(Z) + c, and one sees easily that c = -F(a). We're done. 1R3-10 The table below gives the known values of a function f(s): z 0 1 2 3 4 5 6 f(z) 1 1.2 1.4 1.3 1.5 1.2 1.1 Use Simpson's Rule to estimate the area under the curve y= f(z)between z = 0and z=6. R3-11 Let f(t) be a function, continuous and positive for all t. Let A(s) be the area under the graph of f, between t = 0and t = s. Explain intuitively from the definition of derivative dAwhy =f(s). t <2 R3-12 Let f() = 1 L f(z) O z Evaluate A(s)ds.z-2,2<x<_4 RP-13 Suppose F(s) is a function such that FP(s) -.In terms of …
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