Exercises Limits, Continuity and derivatives (Stewart Ch. 2) Graph 1 Graph 2 Graph 3 Graph 4 See originals 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 The graph of f is shown in graph 1. State the value of each quantity if is exist. If it does not exist explain why. b. c. d. Like exercise 1, now for graph 2 b. c. Like exercise 1, now for graph 3 a. of the function g d. e. For graph 4 of the function f state the following c. d. Sketch the graph of the following function and use it to determine the values of a for which exists d. a. e. a. e. b. f. a. c. g. b. e. The equations of the vertical asymptotes Guess the limit (if it exists) by evaluating the function for x = 2.1, 2.001, 1.9 and 1.99.First check that substituting x = 2 gives the indeterminate form . Guess the limit (if it exists) by evaluating the function for x = 0.5, 0.01 and x = -0.5, -0.01. First check that substituting x = 0 gives you the indeterminate form . a. Evaluating the function for values close to 1. Evaluate the limit and justify each step by indicating the appropriate Limit Law b. Reasoning (function values if x is close to 1) a. What is wrong in the following equation? 2.11 Evaluate the limit if it exists 2.10 2.12 Use the Squeeze Theorem to find the limit. 2.13 2.14 Find the limit if it exists. If not, explain why 2.15 c. 2.16 g(1) g. Sketch the graph of g. Explain why the function is discontinuous at the given number a and sketch the graph of the function: a=2 a=1 a=0 a=0 2.17 State the domain of each function and explain why the function is continuous on this domain. 2.18 Find the numbers at which is discontinuous. At which of these numbers is f continuous from the right or left? 2.19 Use the Intermediate Value Theorem to show that there is a root for the given interval (or find such an interval) , (1,2) 2.20 2.21 Evaluate the limit and justify each step by indicating the appropriate properties of limits 2.22 Find the limit 2.23 2.24 Find the horizontal and vertical asymptotes Find , if for all x > 1: Find the derivative and an equation of the tangent line to the curve at the given point 2.26 If a ball is thrown into the air with a velocity of 10 m/s its height in meters after t seconds is given by: y = 10t – 4.9t2. Find the velocity (the derivative) when t = 2. 2.25 2.27 2.28 Find f `(a) A particle moves along a straight with the equation y = Match the graph of each of the functions (a)-(d) with the graph of its derivatives given in I-IV. 2.30 Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative – x ( y is in m and x in s). Find the velocity at x = 5 2.29 See originals 2.31 True or false quiz: d. if , then e. if does not exist. , then does not exist. f. If exists, then the limit must be f(6)g(6). g. If p is a polynomial then h. If , then =0 i. A function can have to different horizontal asymptotes. j. x=1 is a vertical asymptote of y= f (x) => f(1) is not defined. k. If f(1) > 0 and f(3) < 0, there exists a c such that 1 < x < 3 and f(c)=0 l. If f is continuous at a, then f is differentiable at a.