Math 105D Exam 1 - October 6, 2006, Exams of Calculus

Download Math 105D Exam 1 - October 6, 2006 and more Exams Calculus in PDF only on Docsity! NAME: Math 105D - Exam 1 - October 6, 2006 Instructions: Show all of your work and circle your final answers. Calculators are allowed, but notes and books are not. 1. (25 points) (a) State the limit definition of f ′(a). (b) Briefly explain your definition. What is it intended to calculate? What is the algebraic part calculating? Why is a limit necessary? You may use a graph to assist your explanation, but you must explain what the graph is indicating. (c) Use the limit definition of the derivative (from part a) to show that if f(x) = x − x2, then f ′(3) = −5. 2. (10 points) If h(x) = √ 3x7 +7 √ x− 2 x +4π, find dh dx . (You do not need to use the limit definition here.) 3. (10 points) Consider the second-order differential equation y′′ + y t = t + 2. Is y(t) = t2 a solution? Justify your answer. 4. (16 points) Suppose that the velocity of a particle on the number line at time t is given by v(t) = 6t2−6t. (Positive velocity means the particle is moving to the right, i.e. in the positive direction along the number line.) (a) If the position of the particle at t = 0 is 5, what is the position of the particle at t = 1? (b) On what interval(s) is the particle accelerating to the right?