MIT 18.014 Calculus Practice Exams PDF

PRACTICE EXAM 1 103 √ (1) Compute 99 (2x − 198)2 x − 99 dx where here [x] is defined to be the largest integer ≤ x. (2) Let S be a square pyramid with base area r2 and height h. Using Cavalieri’s Theorem, determine the volume of the pyramid. (3) Let f be...

PRACTICE EXAM 1 103 √ (1) Compute 99 (2x − 198)2 x − 99 dx where here [x] is defined to be the largest integer ≤ x. (2) Let S be a square pyramid with base area r2 and height h. Using Cavalieri’s Theorem, determine the volume of the pyramid. (3) Let f be an integrable function on [0, 1]. Prove that |f | is integrable on [0, 1]. (4) The well ordering principle states that every non-empty subset of the nat­ ural numbers has a least element. Prove the well ordering principle implies the principle of mathematical induction. (Hint: Let S ⊂ P be a set such that 1 ∈ S and if k ∈ S then k + 1 ∈ S. Consider T = P − S. Show that T = ∅.) (5) Suppose limx→p+ f (x) = limx→p− f (x) = A. Prove limx→p f (x) = A. 1 MIT OpenCourseWare http://ocw.mit.edu 18.014 Calculus with Theory Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.  PRACTICE EXAM 2 (1) (10 points) Find 1+h 2 1 2 0 et dt − 0 et dt lim. h→0 h(3 + h2 ) (If you’re using a theorem, state the theorem you’re using.) x (2) (10 points) Find (f −1 ) (0) where f (x) = 0 cos(sin t))dt is defined on [−π/2, π/2]. (3) (10 points) In each case below, assume f is continuous for all x. Find f (2). x f (x) f (t) dt = x2 (1 + x); t2 dt = x2 (1 + x). 0 0 (4) (15 points) Give an example of a function f (x) defined on [−1, 1] such that f is continuous and differentiable on [−1, 1] f is not continuous for at least one value of x ∈ [−1, 1]. (5) (15 points) Let f (x) be continuous on [0, 1] such that f (0) = f (1). Show that for any n ∈ Z+ there exists at least one x ∈ [0, 1] such that f (x) = f (x + 1/n). 1 MIT OpenCourseWare http://ocw.mit.edu 18.014 Calculus with Theory Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.  PRACTICE EXAM 3 3 √t +t dt (1) Evaluate 1+t2 5 √ (2) Evaluate 3 x3 x2 − 9dx (3) Suppose that limx→a+ g(x) = B = 0 where B is finite and limx→a+ h(x) = 0 in a neighborhood of a. Prove that 0, but h(x) = g (x) lim+ = ∞. x→ a h(x) (4) Let f (x) : [0, ∞) → R+ be a positive continuous function such that limx→∞ f (x) = 0. Prove there exists M ∈ R+ such that maxx∈[0,∞) f (x) = M. (5) A sequence is called Cauchy if for all > 0 there exists N ∈ Z+ such that for all m, n > N , |am −an | <. Prove that if {an } is a convergent sequence, then it is Cauchy. (The converse is also true.) A function f : R → R is called a contraction if there exists 0 ≤ α < 1 such that |f (x) − f (y)| ≤ α|x − y|. Let f be a contraction. For any x ∈ R, prove the sequence {f n (x)} is Cauchy, where f n (x) = f ◦f ◦· · ·◦f (x) (the n times composition of f with itself). 1 MIT OpenCourseWare http://ocw.mit.edu 18.014 Calculus with Theory Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.