Set 3: Limits of functions: A. The intuitive approach (.): 1. Watch the video at: https://www.khanacademy.org/math/differential-calculus/it-basics-dc/formal-definition-of-its-dc/v/itintuition-review. 3. 4. Below is the graph of a function f (). For each of the given points determine the value of f (a) and of a. If any of these quantities do not eist, clearly eplain why. 1
5. Below is the graph of a function f (). For each of the given points determine the value of f (a) and of a. If any of these quantities do not eist, clearly eplain why. 6. Below is the graph of a function f (). For each of the given points determine the value of f (a), of a, and of. If any of these quantities do not eist, clearly eplain why. a a
7.. 8. Guess the value of the corresponding it by using the table method. When possible, verify your result algebraically (that is, simplify f () and calculate the it of the simpler epression) or graphically (if you have a graphing device): a) 1 b) c) d) 3 1 e) 4 f) 0 9 3 g) 0 5 e 1 h) sin( ) 0 i) 1 j) 1 1 k) sin 0 3
B. Limit laws (.3): 1. (g) 3 1 g( ) (h) 3 6 h( ). For each of the following eercises, use the it properties to calculate the it. If it is not possible to calculate the it, clearly eplain why not: a) 3 9 4 b) 4 3 c) 4 3 6 3. Evaluate the it, if it eists: a) 4 6 0 3 b) 6 16 4
c) d) e) 4 1 1 3 1 f) 4 1 3 g) 0 1 1 1 1 h) 0 4 i) 1616 4. 5. Find the it, if it eists. If the it does not eist, eplain why: a) 3 3 b) 1 6 6 c) 5
d) 0.5 1 3 e) 1 0 1 6. 7. Given the function: 3 for, evaluate the following its, if they eist: 5 14 for a) 3 and b) 8. 9. 6
C. The Rigorous Definition of Limit (.4) (Optional): 1. Watch the videos at: a) https://www.khanacademy.org/math/differential-calculus/it-basics-dc/formal-definition-of-itsdc/v/building-the-idea-of-epsilon-delta-definition b) https://www.khanacademy.org/math/differential-calculus/it-basics-dc/formal-definition-of-its-dc/v/epsilondelta-definition-of-its and c) https://www.khanacademy.org/math/differential-calculus/it-basics-dc/formal-definition-of-itsdc/v/proving-a-it-using-epsilon-delta-definition. Eplain as precisely (as rigorously) as you can what each of the following means and illustrate with a sketch: 3. Describe several ways in which a it can fail to eist. Illustrate with sketches. 4. Determine whether the following statement is true or false. If it is true, eplain why. If it is false, eplain why or give an eample which disproves the statement: Let f () be a function such that 6 0. Then it is true that there eists a number 0such that if 0 then 6 1. 7
5. What does this graph and your calculations suggest about 1? 6. What does this graph and your calculations suggest about? 3 8
7. What does this graph and your calculations suggest about 1? 8. Prove the following statements using the rigorous - definition of a it: 4 a) 1 3 4 b) 3 5 10 5 c) a d) 0 a 0 e) 71 9
D. Continuity (.5): 1. Write an equation that epresses the fact that a function f () is continuous at the number 4.. If () f is continuous on,, what can you say about its graph? 3. 4. Sketch the graph of a function f () that is continuous ecept at the stated discontinuity: a) discontinuous at -1 and 4, but continuous from the left at -1 and from the right at 4 ; b) has a removable discontinuity at 3 and a non-removable (jump) discontinuity at 5. 5. The graph of a function f () is shown below. Based on this graph, determine where the function is discontinuous: a) 10
b) 6. Use the definition of continuity and the methods to calculate its to determine if the given function is continuous or discontinuous at the given points. Support your work: a) 11
b) c) d) e) 7. Determine where the given function is discontinuous: a) 11 b) 13 7 1 6 3 1
c) 4 1 d) cos 1 1 if 0 if 0 if 8. Find a R such that the function 3 if 3 is continuous for 3. a if 3 9. 10. 11. Use the Intermediate Value theorem to show that there is at least one root of the specified equation in the given interval. Support your work by showing the continuity of the appropriate function: 3 4 a) 1 7 0 on 4,8 b) 11 3 on -15,-5 15 8 c) 0 on -5,1 d) ln 1 ln 4 0 on -1, 3 e) 10 e 5 on 0,4 13
f) e 3 on 0,1 g) sin( ) - on 1, E. Limits at infinity and infinite its. Horizontal Asymptotes and Vertical asymptotes. (.6 and.). 1. Eplain as precisely as you can the meaning of: a) 3 c) 5 b) 4 d) 3. 14
3. 4. 5. Calculate the following infinite its. Support your work: a) 3 3 b) 3 3 c) 3 3 d) 8 5 6 e) 8 5 6 f) 1 3 8 15
6. Find the equations of all the vertical asymptotes for the given function: a) 6 1 b) 9 3 c) sin(4) d) tan(4) 7. Calculate the following its. Support your work: a) 3 1 b) 1 c) 1 1 d) 1 e) 1 1 1 f) cos( ) sin( ) g) cos 4 3 3 1 h) 1 4 i) k) 1 j) 1 l) 1 3 3 m) n) sin ( ) 1 8. Find the horizontal asymptotes of each curve: a) 4 1 1 b) 4 c) 1 3 5 16
F. Derivatives and Rates of Change (.7): 1. Watch the video at:. Solve 10 eercises from: https://www.khanacademy.org/math/differential-calculus/derivative-intro-dc/intro-to-diff-calculus-dc/v/newtonleibniz-and-usain-bolt https://www.khanacademy.org/math/differential-calculus/derivative-intro-dc/intro-to-diff-calculus-dc/e/graphs-offunctions-and-their-derivatives 3. Write an epression for the slope of the tangent line to the curve f () y at the point, f ( a) a. 4. Suppose that an object moves along a straight line with position f (t) at time t. Write an epression for the instantaneous velocity of the object at time t a. How can you interpret this velocity in terms of the graph of f (t)? 5. Find the slope of the tangent line to the parabola line. y 4 at the point (1,3 ). Find an equation of this tangent 6. Find the slope of the tangent line to the curve y 3 at the point (1,0 ). Find an equation of this tangent line. 7. Find an equation of the tangent line to the curve 1 y at the point (1,1). 8. a) Find the slope of the tangent line to the curve y. 3 3 4 at the point where a b) Using part a), find equations of the tangent lines at the points (1,5) and (,3). c) Using a graphical device, graph the curve and both tangents on a common screen. 17
9. 10. a) describe and compare how the runner run the race in terms of their instantaneous speeds. b) at what time is the distance between the runners the greatest? c) at what time do they have the same instantaneous speeds? 18
11. If a ball is thrown into the air with an initial velocity of 40 ft / sec, its height (in feet) after t seconds is given by: y( t) t 40t 16. Find its velocity when t. 1. Find f '( a) for: 3 a) 3 4 1 b) c) G. The derivative as a function (.8): 1. Write a formula for the derivative f '( ) of a function f (). Can you write another formula for f '( )?. 19
3. The graph of a function f () is given. For each function, sketch the graph of its derivative f '( ) directly underneath it: a) b) c) d) 4. Calculate f '( ) using your definition from eercise 1. State the domain of f () and the domain of f '( ) in each case: a) 1 1 b) 3 0
c) 1 d) 1 y 3 5. a) What does it mean for f () to be differentiable at a? b) What is the relationship between differentiability and continuity for a function? c) Sketch the graph of a function which is continuous but not differentiable at a. d) Describe several ways in which a function can fail to be differentiable. Illustrate each case with skecthes. 6. The graph of f () is given. In each case, state with reasons all points at which f () is not differentiable: a) b) c) d) 7. 1