University:
Whitman College
Course:
MATH-125 | Calculus I
Academic year:

2021
Views:

450

Pages:

2
Author:

Cesar Mckenzie

Continuity and Differentiability Worksheet (Be sure that you can also do the graphical exercises from the text- These were not included below! Typical problems are like problems 1-3, p. 161; 1-12, p. 173; 37-38, p. 175; 3,4, p. 133; 44-46, 51 p. 179) 1. Finish the definition: A function f is said to be continuous at x = a if: 2. The definition of continuity implies that we have three things to check. What are they? 3. Finish the definition: A function f is said to be continuous on the interval [a, b] if: 4. Finish the definition: The derivative of f at x = a is: 5. Finish the definition: A function f is said to be differentiable on the interval (a, b) if: 6. Why is the interval open in the last definition? 7. List three interpretations of the derivative of f at x = a. 8. True or False, and give a short reason: (a) If a function is differentiable, then it is continuous. (b) If a function is continuous, then it is differentiable. (c) If f is continuous on [−1, 1] and f (−1) = 4 and f (1) = 3, then there is an x = r so that f (r) = π. (d) If f is continuous at 5, and f (5) = 2, then the limit as x → 2 of f (4x2 − 11) must be 2. (e) All functions are continuous on their domains. 9. Where is each function continuous? r 4 − x2 (a) f (x) = 1 − x2 (b) f (x) = sin−1 (1 − x2 ) x+3 (c) f (x) = ln x−5 x (d) f (x) = 2 x + 5x + 6 10. Explain why the function is discontinuous at the given point, x = a. (a) f (x) = ln |x + 3| at a = −3 (Extra: Is f continuous everywhere else?) (b) f (x) = (c) f (x) = x2 −1 x+1 , x2 −2x−8 , x−4 if x 6= 4 3, if x = 4 a=4 1 − x, if x ≤ 2 x2 − 2x, if x > 2 a=2 at a = −1 (d) f (x) = 11. For each function, determine the value of the constant so that f is continuous everywhere: (a) f (x) = 1 x2 −16 x−4 , if x 6= 4 C, if x = 4 (b)   3x2 − 1, if x < 0 cx + d, if 0 ≤ x ≤ 1 f (x) =  √ x + 8, if x > 1 (c) ( f (x) = √ √ 7x+2− 6x+4 , x−2 if x ≥ − 27 , and x 6= 2 k, if x = 2 12. If f and g are continuous functions with f (3) = 4 and lim [2f (x) − g(x)] = 5, what is g(3)? x→3 13. Show that there must be at least one real solution to x5 − x2 − 4 = 0. 14. Each limit is the derivative of some function at some number a. State f and a in each case: √ 1+h−1 (a) lim h→0 h 9 x −1 (b) lim x→1 x − 1 sin π2 + t − 1 (c) lim t→0 t 15. For each function below, compute the derivative using the definition. Also state the domain of the original function, and the domain of the derivative function. √ (a) f (x) = 1 + 2x (b) g(x) = 1 x2 (c) h(x) = x + (d) f (x) = √ x √2 3−x x x2 −1 (e) f (x) = √ 16. Let f (x) = x. √ √ f (x) − f (a) to compute f 0 (a), for a 6= 0. HINT: x − a = ( x)2 − ( a)2 x−a (b) Show that f 0 (0) does not exist. What does this mean with respect to the graph of f at a = 0? (a) Use lim x→a 17. Given f below, where is f not continuous? f (x) =   0, if x ≤ 0 5 − x, if 0 < x < 4  1 5−x , if x ≥ 4 18. Let f (x) = x3 − 2x. (a) Find f 0 (2). (b) Compute the equation of the line tangent to f at the point (2, 4). 19. Sketch the graph of a function that satisfies the following conditions: g(0) = 0, g 0 (0) = 3, g 0 (1) = 0, g 0 (2) = 1 20. Find the slope of the line tangent to y = x2 + 2x at x = −3, then compute the equation of the (tangent) line. 2

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