Limit & Derivative Problems by ANURAG TYAGI CLASSES (ATC)

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Limit & Derivative Problems by ANURAG TYAGI CLASSES (ATC)

  1. 1. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>1. </li></ul><ul><li>1. </li></ul>
  2. 2. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>2. </li></ul><ul><li>2. </li></ul>
  3. 3. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>3. </li></ul><ul><li>3. </li></ul>
  4. 4. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>4. Consider the function given by </li></ul><ul><li>Is f(x) continuous at x=1? Justify. </li></ul><ul><li>4. </li></ul>
  5. 5. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>5. Is the function given by </li></ul><ul><li>continuous for all x? If not, where are the discontinuities? Are they removable? </li></ul><ul><li>5. </li></ul>
  6. 6. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>6. Let the piecewise function f be defined as follows: </li></ul><ul><li>Which of the following is true about the function f? </li></ul><ul><li>f(2) = 2 </li></ul><ul><li>II. </li></ul><ul><li>f(x) is continuous at x = 2 </li></ul><ul><li>I only </li></ul><ul><li>III only </li></ul><ul><li>I and II only </li></ul><ul><li>I and III only </li></ul><ul><li>I, II, and III </li></ul><ul><li>6. Test: f(2) = 2? Yes, so I is true </li></ul><ul><li>Test: </li></ul><ul><li>Test: f(x) is continuous at x = 2? </li></ul><ul><li>Does the lim f(x) = f(2)? </li></ul><ul><li> 4 is not equal to 2 </li></ul><ul><li> No, so III is false </li></ul><ul><li>Answer is A) I only </li></ul>
  7. 7. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>7. </li></ul><ul><li>What is the value of a for which f(x) is continuous for all values of x? </li></ul><ul><li>-2 </li></ul><ul><li>-1 </li></ul><ul><li>0 </li></ul><ul><li>½ </li></ul><ul><li>1 </li></ul><ul><li>To be continuous at x = 1 </li></ul>
  8. 8. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>Find the cartesian coordinates of the point on the graph of </li></ul><ul><li>where the instantaneous rate of change of f is equal to 5 </li></ul><ul><li>8. </li></ul><ul><li>to find y substitute x = ½ in the original function f(x) </li></ul><ul><li>Ans: (1/2, 11/4) </li></ul>
  9. 9. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>Which of the following directly describes the discontinuities associated with </li></ul><ul><li>a. A hole at x = 3, a vertical asymptote at x = 3 </li></ul><ul><li>b. Holes at x = -3 and x = 3 </li></ul><ul><li>c. A hole at x = 3, a vertical asymptote at x = -3 </li></ul><ul><li>d. Vertical asymptotes at x = 3 and x = -3 </li></ul><ul><li>e. No discontinuities </li></ul><ul><li>9. </li></ul><ul><li>Hole at x = 3 because we factored out (x – 3) </li></ul><ul><li>There is a vertical asymptote at x = -3 </li></ul>
  10. 10. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>Given the piecewise function </li></ul><ul><li>For what values of a and b is f(x) differentiable at x = 1? </li></ul><ul><li>a = 2 b = -3 </li></ul><ul><li>a = 2 b = -2 </li></ul><ul><li>a = -2 b = 1 </li></ul><ul><li>a = 3 b = -1 </li></ul><ul><li>a = 5 b = 8 </li></ul><ul><li>10. Differentiability implies continuity </li></ul><ul><li>To be differentiable x = 1 </li></ul><ul><li>Solve for a when b = 1 </li></ul><ul><li>a – 1 = -3 a = -2 Ans: C </li></ul>
  11. 11. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>Which of the following is (are) true about the function </li></ul><ul><ul><li>It is continuous at x = 0 </li></ul></ul><ul><ul><li>It is differentiable at x = 0 </li></ul></ul><ul><ul><li>I only </li></ul></ul><ul><ul><li>II only </li></ul></ul><ul><ul><li>I and III only </li></ul></ul><ul><ul><li>II and III only </li></ul></ul><ul><ul><li>I, II, III </li></ul></ul><ul><li>Test 1 : Continuous at x = 0 </li></ul><ul><li>yes </li></ul><ul><li>Test 2 : Differentiable at x = 0? </li></ul><ul><li>No </li></ul><ul><li>Test 3: </li></ul><ul><li>Yes </li></ul><ul><li>Ans: C </li></ul>
  12. 12. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>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? </li></ul><ul><li>f is continuous on (a, b) and differentiable on (a, b) </li></ul><ul><li>f is continuous on (a, b) and differentiable on [a, b] </li></ul><ul><li>f is continuous on (a, b) and differentiable on [a, b) </li></ul><ul><li>f is continuous on [a, b] and differentiable on (a, b) </li></ul><ul><li>f is continuous on [a, b] and differentiable on [a, b] </li></ul><ul><li>Look at the definition of Rolle’s Theorem and the Mean Value Theorem </li></ul><ul><li>f is continuous on [a, b] and differentiable on (a, b) </li></ul><ul><li>Ans: D </li></ul>
  13. 13. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>Let f be the function defined by </li></ul><ul><li>A. Determine the x and y intercepts, if any. Justify your answer. </li></ul><ul><li>A </li></ul>
  14. 14. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>Let f be the function defined by </li></ul><ul><li>B. Write an equation for each vertical and each horizontal asymptote. Justify your answer. </li></ul><ul><li>B </li></ul><ul><li>Vertical asymptote </li></ul><ul><li>Horizontal asymptote </li></ul>
  15. 15. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>Let f be the function defined by </li></ul><ul><li>C. Determine the intervals on which f is increasing or decreasing. Justify your answer. </li></ul><ul><li>C </li></ul>
  16. 16. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>Let f be the function defined by </li></ul><ul><li>D. Determine the relative minimum and maximum points, if any. Justify your answer. </li></ul><ul><li>D </li></ul><ul><li>Relative minimum occurs at x = -2 </li></ul><ul><li>when x = -2 </li></ul><ul><li> </li></ul>
  17. 17. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>Let f be the function defined by </li></ul><ul><li>E. Determine the intervals on which f is concave up or concave down. Justify your answer. </li></ul><ul><li>E </li></ul>
  18. 18. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>Let f be the function defined by </li></ul><ul><li>F. Determine any points of inflection </li></ul><ul><li>F </li></ul><ul><li>Point of inflection when x = 3 </li></ul>
  19. 19. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>On the interval [1, 3], what is the average rate of change for the functions, if </li></ul><ul><li>14. </li></ul>
  20. 20. Limit & Derivative Problems <ul><li>Problem… </li></ul><ul><li>Answer and Work… </li></ul><ul><li>Is the function defined by </li></ul><ul><li>continuous at x = 4? Justify your answer. </li></ul><ul><li>15. </li></ul>

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