PRACTICE EXAM FOR FIRST MIDTERM EXAMANDREW J. BLUMBERG1. NotesThis practice exam has more problems than the real exam will. To make mosteffective use of this document, take the exam under conditions simulating the realexam — no book, no calculator.(1) Short answer questions:(a) Do all functions that have an inverse have a derivative?(b) Give an example of a function which is continuous but not differen-tiable.(c) Consider the function f(x) defined to be 0 if x ≤ 0 and 1 otherwise.Although f (−1) = 0 and f(2) = 1, f never takes on the value 0.5.Why doesn’t this contradict the intermediate value theorem?(d) Explain how to use continuity to evaluate limits.(e) Explain the horizontal line test.(2) Solve the following equations for x.(a)53x= 25x2−x+2.(b)22x−2= 7x2(c)e2x− 4ex+ 3 = 0.(3) Find the inverse of x4on the domain [0, 64], on the domain [−64, 64], andon the domain [−64, 0].(4) Find the inverses of the following functions algebraically:(a) f(x) = 5 − e−x.(b) f(x) =1+x1−x.(Please explain any restrictions on the domain or range.)(5) Sketch the graph of the inverse of the function y = ln(x − 3).(6) The position of a particle at time t is given by f (t) = t3− t + 1.(a) Find the average velocity over the interval [0, 2].(b) Find the derivative using the definition, and find the average acceler-ation over the same interval.(7) Describe the behavior of the slope of the tangents to sin x as x varies over[0, 2π].(8) Draw a graph of the position and a separate graph of the velocity of yourcar as you drive across the country. (Please assume realistic conditions;e.g., you need to sleep occasionally.)Date: September 17, 2011.12 ANDREW J. BLUMBERG(9) Suppose that limx→1(f(x))2= 3. What is limx→1f(x)? (Does it have toexist?)(10) Prove formally that limx−>∞3exis ∞.(11) Find the following limits:(a)limx→−∞x5+ 3(x15+ 3x9− 4x6+ 2)13.(b)limx→3x2− x − 6x − 3.(12) Use the intermediate value theorem to:(a) Locate a solution for the equation:log2(x + 1) + log4(x − 1) = 1.(b) Show that x4= −1 has no solutions.(13) (a) Consider the following procedure: take a positive number x, rounddown to an integer, take the remainder when you divide by 12, add1, and find the number of days in that month. Does this describe acontinuous function?(b) Where is f (x) = esin(x)+x−10√xcontinuous?(14) Let f (x) =(x+3)3(x−4)(x−1)(x−2). Sketch the horizontal and vertical asymptotesof f (x).(15) Are the slopes of the tangents of f (x) = x2+ 2 and g(x) = 3x3− 4x + 1ever parallel? (Please use the definition to find the derivatives in order todo this.)E-mail address:
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