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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 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 R10 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)a c)R= + =? RI1 Differentiate: a) sin b) sin 2 (vr) c)Xz/Stan d) 2 + 2 e) c( 1) f) cos 3 (v ) g)tan( s3 ) h) sec 2 (3s + 1) R12 Consider f () = 2z 2 + 4z + 3. Where does the tangent line to the graph of f(z) at x =3 cross the yaxis? R13 Find the equation of the tangent to the curve 2s 2 + zy y + 2x3y =20 at the point (3,2). R14 Define the derivative of f(x) . Directly from the definition, show that f'(z) = cos if f(x) = sin .(Youmay use without proof: lim sn = , lim coh1 0). h+o h h+O h R15 Find all real so such that f'(xo) = 0: a) f(s) = X + 1 b) f () = + cosa R16 At what points is the tangent to the curve y.+ Xy X + 3 = 0 horizontal? R17 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. R18 Derive a formula for (xl)' . R19 a) What is the rate of change of the area A of a square with respect to its side x ? b) What is the rate of change of the area A of a circle 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 RI10 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. R111 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 Rl12 Find all values of the constants a and b so that the function f() = az+b, z > ir will be (a) continuous; (b) differentiable. R113 Evaluate li sin( 4 z) (Hint: Let 4z = t.) 2+0 Z Unit 2: Applications of Differentiation R21 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 2 z, 0 < s < 2r c) f () = Z + 1/z 2 d) f(z) = z +sin 2z R22 A baseball diamond is a 90 ft. square. A ball 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 ?...
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 Spring '11
 Blake
 Math, Calculus, Derivative, Continuous function

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