First Midterm Exam: Math 125 - Calculus I Fall 2017 September 11, 2017 Name (please print): Signature: Student ID: Instructions to Candidates 1. This examination paper contains SEVEN (7) questions and comprises EIGHT (8) printed pages. 2. Answer ALL the questions in the space provided. The points for each question are indicated at the beginning of each question. 3. This IS NOT an OPEN BOOK exam. 4. Approved calculators are permitted. However, you should write down systematically the steps in the workings. Also the answers should always be in exact and simplified form unless you are specifically asked to estimate. 5. Answers with no explanation will not receive full credit. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Total This page is intentionally left blank. 2 Question 1. [20 points] Find whether the following statements are True or False. Circle your answer. (2 points each) (i) TRUE / FALSE If the function f is invertible, then the equation y = f 1 (x) of the graph of the inverse function is defined by the relation x = f (y). (ii) TRUE / FALSE The straight line x+2y = 2 is perpendicular to the straight line 2x+y = 1. (iii) TRUE / FALSE If f (x) = ex and g(x) = x2 1, then lim f (g(x)) = 1. (iv) TRUE / FALSE If lim+ f (x) = 1 and lim f (x) = 1, then f (0) = 1. x1 x0 x0 (v) TRUE / FALSE If f (x) and g(x) are continuous functions, then f (g(x)) is also continuous. (vi) TRUE / FALSE If f (1) = 0 and f (x) is continuous at x = 1, then lim f (x) = 0 and x1 lim+ f (x) = 0. x1 (vii) TRUE / FALSE If f 0 (x) > 0 then f (x) is decreasing. (viii) TRUE / FALSE If f (x) and g(x) are odd functions, then the product function f (x)g(x) is also an odd function. (ix) TRUE / FALSE (x) TRUE / FALSE lim f (x) = x 2x2 3x + 5 = 2. x2 + x + 1 f (a) f (a h) . h0 h The derivative of f at a is f 0 (a) = lim 3 Question 2. [15 points] Evaluate the following limits without using any method involving the differentiation, for example L'Hopital's rule. x2 + 2x 3 . (8 points) (i) lim x1 x1 e3x + 6 . x 2 ex (ii) lim (7 points) 4 Question 3. Find the value of k such that the following function f is continuous everywhere. ( ex , x0 f (x) = (x + k), x < 0. [10 points] Question 4. [20 points] Suppose a particle is moving along a straight line. At time t in seconds, the particle's distance s(t) in meters, from a certain fixed point is given by s(t) = 5t2 . (i) Find the average velocity over the time interval 1 t 1.1 in appropriate units. (3 points) 5 (ii) Find the average velocity over the time interval 1 t 1.01 in appropriate units. points) (4 (iii) Use the definition to find the instantaneous velocity at t = 1 in appropriate units. points) (7 (iv) Use the answer in part (iii) to estimate s(1.002). 6 (6 points) Question 5. [12 points] 1 Find the derivative function f 0 (x) for f (x) = by using the definition. 2x + 1 Question 6. [8 points] Sketch a possible graph of y = f (x) given the following three conditions about the derivative: f 0 (x) < 0 for 1 < x < 1, f 0 (x) > 0 for x > 1 and x < 1, and f 0 (x) = 0 for x = 1 and x = 1. 7 Question 7. Consider the function g defined by [15 points] g(x) = x + x2 . (i) Use the definition to find the derivative of g(x) at x = 1. (8 points) (ii) Use the above answer to find the equation of the tangent line to the graph y = g(x) at x = 1. (7 points) END OF PAPER 8