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HARVARD MATH 1A - First hourly Practice

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Math 1A: introduction to functions and calculus Oliver Knill, Spring 20113/1/2011: First hourly PracticeYour Name:• Start by writing your name in the above b ox.• Try to answer each question on t he same page as the question is asked. If needed, use theback or the next empty page for work. If you need additional paper, write your name on it.• Do not detach pages from this exam packet or unstaple the packet.• Please write neatly and except for multiple choice problems, give computations. Answerswhich are illegible for the grader can not be given credit.• No notes, books, calculators, computers, or other electronic aids can be allowed.• You have 90 minutes time to complete your work.• All unspecified functions are assumed to be smooth: one can differentiate arbitrarily.• The actual exam has a similar format: TF questions, multiple choice and then problemswhere work needs to be shown.1 202 103 104 105 106 107 108 109 1010 10Total: 110Problem 1) True/False questions (20 points) No justifications are needed.1)T FThe function cot(x) is the inverse of the function tan(x).2)T FWe have cos(x)/ sin(x) = cot(x)3)T Fsin(3π/2) = −1.4)T FThe function f(x) = sin(x)/x has a limit at x = 0.5)T FFor the function f(x) = sin(sin(exp(x))) the limit limh→0[f(x+h)−f(x)]/hexists.6)T FIf a differentiable function f (x) satisfies f′(3) = 3 and is f′is odd then ithas a critical point.7)T FThe l’Hopital rule assures that the derivative satisfies (f /g)′= f′/g′.8)T FThe intermediate value theorem assures that a continuous function has aderivative.9)T FThe function f(x) = (x + 1)/(x2− 1) is continuous everywhere.10)T FIf f is concave up on [1, 2] and concave down on [2, 3] then 2 is an inflectionpoint.11)T FThere is a function f which has the property that its second derivative f′′is equal to its negative f .12)T FThe function f (x) = [x]4= x(x + h)(x + 2h)(x + 3h) has the property thatD f (x) = 4[x]3= 4x(x + h)(x + 2h), where Df(x) = [f(x + h) − f (x)]/h.13)T FThe quotient rule is d/dx(f/g) = (f′g −fg′)/g2and holds whenever g(x) 6=0.14)T FThe chain rule assures that d/ dxf(g(x)) = f′(g(x)) + f(g′(x)).15)T FIf f and g are differentiable, then (3f + g)′= 3f′+ g′.16)T FFor any f unction f, the Newton step T (x) is continuous.17)T FOne can rotate a four legged table on an arbitrary surface such that all fourlegs are on the ground.18)T FThe fundamental theorem of calculus relates integration S with differentia-tion D. The r esult is DSf(x) = f (x), SDf(x) = f(x) − f (0).19)T FThe product rule implies d/dx(f(x)g(x)h(x)) = f′(x)g(x)h(x) +f(x)g′(x)h(x) + f (x)g(x)h′(x).20)T FEuler and Gauss are the founders of infinitesimal calculus.Problem 2) Matching problem (10 points) No justifications are needed.Match the following functions with their gra phs.Function Fill in 1-8x2− xexp(−x)sin(3x)log(|x|)tan(x)1/(2 + cos(x))x − cos(6x)sin(3x)/x1) 2) 3) 4)5) 6) 7) 8)Problem 3) Matching problem (10 points) No justifications are needed.Match the following functions with their derivatives.Function Fill in the numbers 1-8graph a)graph b)graph c)graph d)graph e)graph f)graph g)graph h)a)b) c) d)e) f) g) h)1) 2) 3) 4)5) 6) 7) 8)Problem 4) Functions (10 points) No justifications are neededMatch the following functions. In each of the cases, exactly one o f the choices A-C is true.Function Choice A Choice B Choice C Enter A-Cx4−1x−11 + x + x2+ x31 + x + x21 + x + x2+ x3+ x42xe2 log(x)ex log(2)2e log(x)sin(2x) 2 sin(x) cos(x) cos2(x) − sin2(x) 2 sin(x)1/x + 1/(2x) 1/(x + 2x) 3x/2 1/(x + 2x)ex+2exe22ex(ex)2log(4x) 4 log(x) log(4) log(x) log(x) + log(4)√x3x3/2x2/33√xProblem 5) Roots (10 points)Find the roots of the following functionsa) (2 points) 7 sin(3πx)b) (2 points) x5− x.c) (2 points) log |ex|.d) (2 points) e5x− 1e) (2 points) 8x/(x2+ 4) − x.Problem 6) Derivatives (10 points)Find the derivatives of the following functions:a) (2 points) f(x) = cos(3x)/ cos(10x)b) (2 points) f(x) = sin2(x) log(1 + x2)c) (2 points) f(x) = 5x4− 1/(x2+ 1)d) (2 points) f(x) = tan(x) + 2xe) (2 points) f(x) = arccos(x)Problem 7) Limits (10 points)Find the limits limx→0f(x) of the following functions:a) (2 points) f(x) = (x6− 3x2+ 2x)/(1 + x2− cos(x)).b) (2 points) f(x) = (cos(3x) − 1) /(cos(7x) − 1).c) (2 points) f(x) = tan3(x)/x3.d) (2 points) f(x) = sin(x) log(x6)e) (2 points) f(x) = 4x(1 − x)/(cos(x) − 1).Problem 8) Extrema (10 points)a) (5 points) Find all local extrema of the function f (x) = 30x2− 5x3− 15x4+ 3x5on the realline.b) (5 points) Find the global maximum and global minimum of the function f (x) = exp(x) −exp(2x) on the interval [−2, 2].Problem 9) Extrema (10 points)A cup o f height h and radius r has the volume V =πr2h. Its surface area is πr2+ πrh. Among all cupswith volume V = π find the one which has minimalsurface area. Find the global minimum.Problem 10) Newton method (10 points)a) (3 points) Produce the first Newton step for the function f(x) = ex− x at the point x = 1.b) (4 points) Produce a second Newton step.c) (3 points) Find the Newton step map T (x) if the function f(x) is replaced by the


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