Limit & Derivative Problems by ANURAG TYAGI CLASSES (ATC)
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The document covers various limit and derivative problems involving continuity, differentiability, and analyzing functions. It addresses questions about piecewise functions, continuity at specific points, and the application of the Mean Value and Rolle’s theorems. Key topics include finding coordinates, determining discontinuities, and justifying results for function behavior.
![Limit & Derivative Problems Problem… Answer and Work… To apply either the Mean Value Theorem or Rolle’s Theorem to a function f, certain requirements regarding the continuity and differentiability of the function must be met. Which of the following states the requirements correctly? f is continuous on (a, b) and differentiable on (a, b) f is continuous on (a, b) and differentiable on [a, b] f is continuous on (a, b) and differentiable on [a, b) f is continuous on [a, b] and differentiable on (a, b) f is continuous on [a, b] and differentiable on [a, b] Look at the definition of Rolle’s Theorem and the Mean Value Theorem f is continuous on [a, b] and differentiable on (a, b) Ans: D](https://image.slidesharecdn.com/limitderivativeproblems-100125100048-phpapp01/85/Limit-Derivative-Problems-by-ANURAG-TYAGI-CLASSES-ATC-12-320.jpg)
![Limit & Derivative Problems Problem… Answer and Work… On the interval [1, 3], what is the average rate of change for the functions, if 14.](https://image.slidesharecdn.com/limitderivativeproblems-100125100048-phpapp01/85/Limit-Derivative-Problems-by-ANURAG-TYAGI-CLASSES-ATC-19-320.jpg)
Limit & Derivative Problems by ANURAG TYAGI CLASSES (ATC)
- 1. Limit & Derivative Problems Problem… Answer and Work… 1. 1.
- 2. Limit & Derivative Problems Problem… Answer and Work… 2. 2.
- 3. Limit & Derivative Problems Problem… Answer and Work… 3. 3.
- 4. Limit & Derivative Problems Problem… Answer and Work… 4. Consider the function given by Is f(x) continuous at x=1? Justify. 4.
- 5. Limit & Derivative Problems Problem… Answer and Work… 5. Is the function given by continuous for all x? If not, where are the discontinuities? Are they removable? 5.
- 6. Limit & Derivative Problems Problem… Answer and Work… 6. Let the piecewise function f be defined as follows: Which of the following is true about the function f? f(2) = 2 II. f(x) is continuous at x = 2 I only III only I and II only I and III only I, II, and III 6. Test: f(2) = 2? Yes, so I is true Test: Test: f(x) is continuous at x = 2? Does the lim f(x) = f(2)? 4 is not equal to 2 No, so III is false Answer is A) I only
- 7. Limit & Derivative Problems Problem… Answer and Work… 7. What is the value of a for which f(x) is continuous for all values of x? -2 -1 0 ½ 1 To be continuous at x = 1
- 8. Limit & Derivative Problems Problem… Answer and Work… Find the cartesian coordinates of the point on the graph of where the instantaneous rate of change of f is equal to 5 8. to find y substitute x = ½ in the original function f(x) Ans: (1/2, 11/4)
- 9. Limit & Derivative Problems Problem… Answer and Work… Which of the following directly describes the discontinuities associated with a. A hole at x = 3, a vertical asymptote at x = 3 b. Holes at x = -3 and x = 3 c. A hole at x = 3, a vertical asymptote at x = -3 d. Vertical asymptotes at x = 3 and x = -3 e. No discontinuities 9. Hole at x = 3 because we factored out (x – 3) There is a vertical asymptote at x = -3
- 10. Limit & Derivative Problems Problem… Answer and Work… Given the piecewise function For what values of a and b is f(x) differentiable at x = 1? a = 2 b = -3 a = 2 b = -2 a = -2 b = 1 a = 3 b = -1 a = 5 b = 8 10. Differentiability implies continuity To be differentiable x = 1 Solve for a when b = 1 a – 1 = -3 a = -2 Ans: C
- 11. Limit & Derivative Problems Problem… Answer and Work… Which of the following is (are) true about the function It is continuous at x = 0 It is differentiable at x = 0 I only II only I and III only II and III only I, II, III Test 1 : Continuous at x = 0 yes Test 2 : Differentiable at x = 0? No Test 3: Yes Ans: C
- 12. Limit & Derivative Problems Problem… Answer and Work… To apply either the Mean Value Theorem or Rolle’s Theorem to a function f, certain requirements regarding the continuity and differentiability of the function must be met. Which of the following states the requirements correctly? f is continuous on (a, b) and differentiable on (a, b) f is continuous on (a, b) and differentiable on [a, b] f is continuous on (a, b) and differentiable on [a, b) f is continuous on [a, b] and differentiable on (a, b) f is continuous on [a, b] and differentiable on [a, b] Look at the definition of Rolle’s Theorem and the Mean Value Theorem f is continuous on [a, b] and differentiable on (a, b) Ans: D
- 13. Limit & Derivative Problems Problem… Answer and Work… Let f be the function defined by A. Determine the x and y intercepts, if any. Justify your answer. A
- 14. Limit & Derivative Problems Problem… Answer and Work… Let f be the function defined by B. Write an equation for each vertical and each horizontal asymptote. Justify your answer. B Vertical asymptote Horizontal asymptote
- 15. Limit & Derivative Problems Problem… Answer and Work… Let f be the function defined by C. Determine the intervals on which f is increasing or decreasing. Justify your answer. C
- 16. Limit & Derivative Problems Problem… Answer and Work… Let f be the function defined by D. Determine the relative minimum and maximum points, if any. Justify your answer. D Relative minimum occurs at x = -2 when x = -2
- 17. Limit & Derivative Problems Problem… Answer and Work… Let f be the function defined by E. Determine the intervals on which f is concave up or concave down. Justify your answer. E
- 18. Limit & Derivative Problems Problem… Answer and Work… Let f be the function defined by F. Determine any points of inflection F Point of inflection when x = 3
- 19. Limit & Derivative Problems Problem… Answer and Work… On the interval [1, 3], what is the average rate of change for the functions, if 14.
- 20. Limit & Derivative Problems Problem… Answer and Work… Is the function defined by continuous at x = 4? Justify your answer. 15.