8 Questions on Calculus I - Exam 1 | MATH 1910, Exams of Calculus

Download 8 Questions on Calculus I - Exam 1 | MATH 1910 and more Exams Calculus in PDF only on Docsity! Math 1910 Exam 1 - Spring 2006 (55 mins) Name: Note: Use the paper provided to construct your solutions. This is a closed notes, closed book test. You may not use a calculator. Show all work to maximize any partial credit. 1. Write an equation that expresses the fact that a function f is continuous at the number 4. (10) 2. Calculate the limits if they exist. If the limit is ∞ or −∞, so indicate. (10 ea) a. lim x→1 arcsin ( 1− √ x 1− x ) b. lim u→∞ 4u4 + 5 (u2 − 2)(2u2 − 1) c. lim t→4 t2 − 16 t + 2 d. lim x→1− x2 − 1 |x− 1| 3. Where are each of the functions continuous? Explain your conclusions. (5 ea) h(θ) = ln(1 + cos θ) p(x) = x2 + √ 2x− 1 4. Find a formula for the inverse of the function f . What property of f ensures that f−1 exists? f(x) = ex 3−1 (10) 5. Show, with explanation included, whether the following equation has at least one real root on the interval (0, π/2). (10) sinx = 1− x 6. Let f(x) = √ 1 + 2x. (5 ea) • State the domain of f . • Find the derivative of f using the definition of the derivative. • State the domain of f ′. • Find the equation of the tangent line at the point P (3/2, 2). Extra Credit: (Note: No partial credit for extra credit problems.) (2 pts each) 1. Plot f(x) = ln(x− 1) along with its derivative (on the same axis). Label each accordingly. 2. Plot f and f ′ on the same axis, labelling each accordingly, where f(x) = { |x| : −1 ≤ x ≤ 1 −|x− 2|+ 2 : 1 < x ≤ 3.