Chapter 4.1 & 4.2 (Part 1) Practice Problems
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- Gordon Sherman
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1 Chapter 4. & 4. Part Practice Problems EXPECTED SKILLS: Understand how the signs of the first and second derivatives of a function are related to the behavior of the function. Know how to use the first and second derivatives of a function to find intervals on which the function is increasing, decreasing, concave up, and concave down. Be able to find the critical points of a function, and apply the First Derivative Test and Second Derivative Test when appropriate to determine if the critical points are relative maxima, relative minima, or neither Know how to find the locations of inflection points. PRACTICE PROBLEMS:. Consider the graph of y = fx, shown below. a Determine the intervals where fx is increasing. b, d f, b Determine the intervals where fx is decreasing., b d, f c Determine the intervals where fx is concave up., c e,
2 d Determine the intervals where fx is concave down. c, e e Determine the values of x where fx has relative local extrema. Classify each as the location of a relative maximum or a relative minumum. Relative max when x = d; Relative minima when x = b and x = f f Determine the values of x where fx has an inflection point. Point of Intersection when x = c and x = e. The graph of the derivative of y = fx is shown below. a Determine the intervals where fx is increasing., a g, b Determine the intervals where fx is decreasing. a, d d, g c Determine the intervals where fx is concave up. b, d f, d Determine the intervals where fx is concave down., b d, f e Determine the values of x where fx has relative local extrema. Classify each as the location of a relative maximum or a relative minumum. Relative maximum when x = a; Relative minimum when x = g; Neither a relative max nor a relative min at the critical point of x = d.
3 f Determine the values of x where fx has an inflection point. Points of inflection when x = b, x = d and x = f 3. Sketch the graph of a continuous function, y = fx, which is decreasing on,, has an inflection point at x =, and is concave down on,. 4. Sketch the graph of a continuous function, y = fx, which is decreasing on,, has a relative minimum at x =, and does not have any inflection points. or 5. Sketch the graph of a continuous function y = fx which satisfies all of the following conditions: Domain of fx is, f =, f0 = f7 = 3, and f5 = 9 f x < 0 on, 5, 7 and f x > 0 on, 0 0, 5 7, f x < 0 on 0, 7 7, and f x > 0 on, 0 3
4 6. Consider the function that you sketched in question 5. At which values of x must f x = 0? At which values of x must f x fail to exist? f x = 0 when x = and x = 5; f x DNE when x = 7 For problems 7-5, calculate each of the following: a The intervals on which fx is increasing b The intervals on which fx is decreasing c The intervals on which fx is concave up d The intervals on which fx is concave down e All points of inflection. Express each as an ordered pair x, y 7. fx = x 3 x + 3 a., 3 8. fx = x x 3, ; b. 3, a. none; b.,, ; c., ; d., ; e. none ; c. 0, ; d., 0; e. 0, fx = sin x on [0, π] [ a. 0, π 3π, π ; b. π, 3π ; c. π, π; d. 0, π; e. π, 0 0. fx = 4x 4 a. 4, ; b., ; c. 4, 4 4, ; d. none; e. none. fx = xe x a., ; b., ; c., ; d., ; e., e. fx = arctan x a., ; b. none; c., 0; d. 0, ; e. 0, 0 3. fx = π e x / a., 0; b.0, ; c.,, ; d., ; e., and, πe / πe / 4
5 4. fx = ln x x a. 0, e; b. e, ; c. e 3/, ; d. 0, e 3/ ; e. e 3/ 3, e 3/ 5. fx = x + 3x /3 a., 0, ; b., 0; c. none; d., 0 0, ; e. none For problems 6-0, compute the critical points of the given function. Then use the First Derivative Test to determine all relative local extrema. Express each extremum as an ordered pair x, y. 6. fx = x 6 Relative min at 0, 6 7. fx = x Critical Point at 3, No relative extrema 8. fx = 3x x + Relative max at, 3 ; Relative min at, 3 9. fx = e x x Relative min at 0, 0. fx = x 3 x 5 3/ 3 3 Relative maximum at 5, 5 5 3/ 3 3 Relative minimum at 5, 5 5 Critical point at 0, 0, which is neither a relative max nor a relative min For problems -, use the Second Derivative Test to determine the relative local extrema. Express each as an ordered pair x, y.. fx = sin 3x on [0, π] π Relative maxima at 6, and 5π π 6, ; Relative minimum at, 5
6 . fx = sec 3x on [0, π] Relative minima at 0, and π π 3, ; Relative maxima at 3, and π, For problems 3-7, determine the critical points. Classify each as a relative extremum, relative minimum, or neither. Express all relative extrema as ordered pairs x, y. 3. fx = sin x on [0, π] Relative minima at 0, 0, π, 0, and π, 0; π 3π Relative maxima at,, and, 4. fx = x3 3 + x + x + 3 No relative extrema 5. fx = xe x Relative minimum at, e 6. fx = x + 3x /3 Relative maximum at, ; Relative minimum at 0, 0 7. fx = ln x x Relative Maximum at e, e HINT: For problems 5-7, it may be helpful to use your work from earlier in the assignment. 6
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