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Mathematics LibreTexts

3.E: Applications of the Graphical Behavior of Functions(Exercises)

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    9971
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    3.1: Extreme Values

    Terms and Concepts

    1. Describe what an "extreme value" of a function is in your own words.

    2. Sketch the graph of a function \(f\) on (-1,1) that has both a maximum and minimum value.

    3. Describe the difference between absolute and relative maxima in your own words.

    4. Sketch the graph of a function \(f\) where \(f\) has a relative maximum at \(x=1\) and \(f'(1)\) is undefined.

    5. T/F: If \(c\) is a critical value of a function \(f\), then \(f\) has either a relative maximum or relative minimum at \(x=c\).

    Problems

    In Exercises 6-7, identify each of the marked points as being an absolute maximum and minimum, a relative maximum or minimum, or none of the above.

    6.
    3106.PNG

    7.
    3107.PNG

    In Exercises 8-14, evaluate \(f'(x)\) at the points indicated in the graph.

    8. \(f(x)=\frac{2}{x^2+1}\)
    3108.PNG

    9. \(f(x) = x^2\sqrt{6-x^2}\)
    3109.PNG

    10. \(f(x)=\sin x\)
    3110.PNG

    11. \(f(x) = x^2\sqrt{4-x}\)
    3111.PNG

    12. \(f(x) =\begin{cases} x^2 \quad &x\le 0 \\ x^5 &x>0 \end{cases}\)
    3112.PNG

    13. \(f(x) =\begin{cases} x^2 \quad &x\le 0 \\ x &x>0 \end{cases}\)
    3113.PNG

    14. \(f(x) = \frac{(x-2)^{2/3}}{x}\)
    3114.PNG

    In Exercises 15-24, find the extreme values of the function on the given interval.

    15. \(f(x) =x^2+x+4\text{ on }[-1,2]\).

    16. \(f(x) =x^3-\frac{9}{2}x^2-30x+3\text{ on }[0,6]\).

    17. \(f(x) =3\sin x\text{ on }[\pi/4,2\pi/3]\).

    18. \(f(x) =x^2\sqrt{4-x^2}\text{ on }[-2,2]\).

    19. \(f(x) =x+\frac{3}{x}\text{ on }[1,5]\).

    20. \(f(x) =\frac{x^2}{x^2+5}\text{ on }[-3,5]\).

    21. \(f(x) =e^x\cos x\text{ on }[0,\pi]\).

    22. \(f(x) =e^x\sin x\text{ on }[0,\pi]\).

    23. \(f(x) =\frac{\ln x}{x}\text{ on }[1,4]\).

    24. \(f(x) =x^{2/3}-x\text{ on }[0,2]\).

    Review

    25. Find \(\frac{dy}{dx}\), where \(x^2y-y^2x=1\).

    26. Find the equation of the line tangent to the graph of \(x^2+y^2+xy=7\) at the point \((1,2)\).

    27. Let \(f(x)=x^3+x\). Evaluate \(\lim\limits_{s\to 0} \frac{f(x+s)-f(x)}{s}\).

    3.2: The Mean Value Theorem

    Terms and Concepts

    1. Explain in your own words what the Mean Value Theorem states.

    2. Explain in your own words what Rolle's Theorem states.

    Problems

    In Exercises 3-10, a function \(f(x)\) and interval [a,b] are given. Check if Rolle's Theorem can be applied to \(f\) on [a,b]; if so, find \(c\) in [a,b] such that \(f'(c)=0\).

    3. \(f(x) =6\text{ on }[-1,1]\).

    4. \(f(x) =6x\text{ on }[-1,1]\).

    5. \(f(x) =x^2+x-6\text{ on }[-3,2]\).

    6. \(f(x) =x^2+x-2\text{ on }[-3,2]\).

    7. \(f(x) =x^2+x\text{ on }[-2,2]\).

    8. \(f(x) =\sin x \text{ on }[\pi/6,5\pi/6]\).

    9. \(f(x) =\cos x\text{ on }[0,\pi]\).

    10. \(f(x) =\frac{1}{x^2-2x+1}\text{ on }[0,2]\).

    In Exercises 11-20, a function \(f(x)\) and interval [a,b] are given. Check if the Mean Value Theorem can be applied to \(f\) on [a,b]; if so, find a value \(c\) in [a,b] guaranteed by the Mean Value Theorem.

    11. \(f(x) =x^2+3x-1\text{ on }[-2,2]\).

    12. \(f(x) =5x^2-6x+8\text{ on }[0,5]\).

    13. \(f(x) =\sqrt{9-x^2}\text{ on }[0,3]\).

    14. \(f(x) =\sqrt{25-x}\text{ on }[0,9]\).

    16. \(f(x) =\ln x\text{ on }[1,5]\).

    17. \(f(x) =\tan x\text{ on }[\pi/4, \pi/4]\).

    18. \(f(x) =x^3-2x^2+x+1\text{ on }[-2,2]\).

    19. \(f(x) =2x^3-5x^2+6x+1\text{ on }[-5,2]\).

    20. \(f(x) =\sin^{-1}x\text{ on }[-1,1]\).

    Review

    21. Find the extreme values of \(f(x)=x^2-3x+9\text{ on }[-2,5]\).

    22. Describe the critical points of \(f(x) =\cos x\).

    23. Describe the critical points of \(f(x)=\tan x\).

    3.3: Increasing and Decreasing Functions

    Terms and Concepts

    1. In your own words describe what it means for a function to be increasing.

    2. What does a decreasing function “look like”?

    3. Sketch a graph of a function on [0,2] that is increasing but not strictly increasing.

    4. Give an example of a function describing a situation where it is “bad” to be increasing and “good” to be decreasing.

    5. A function f has derivative \(f ′ (x) = (\sin x + 2)e^{x^2+1}\), where \(f ′ (x) > 1\) for all \(x\). Is \(f\) increasing, decreasing, or can we not tell from the given information?

    Problems

    In Exercises 6-13, a function \(f(x)\) is given.
    (a) Compute \(f'(x)\).
    (b) Graph \(f\) and \(f'\) on the same axes (using technology is permitted) and verify Theorem 29.

    6. \(f(x) =3x+4\)

    7. \(f(x) =x^2-3x+5\)

    8. \(f(x) =\cos x\)

    9. \(f(x) =\tan x\)

    10. \(f(x) =x^3-5x^2+7x-1\)

    11. \(f(x) =2x^3-x^2+x-1\)

    12. \(f(x) =x^4-5x^2+4\)

    13. \(f(x) =\frac{1}{x^2+1}\)

    In Exercises 14-23, a function \(f(x)\) is given.
    (a) Give the domain of \(f\).
    (b) Find the critical numbers of \(f\).
    (c) Create a number line to determine the intervals on which \(f\) is increasing and decreasing.
    (d) Use the First Derivative Test to determine whether each critical point is a relative maximum, minimum, or neither.

    14. \(f(x) =x^2+2x-3\)

    15. \(f(x) =x^3+3x^2+3\)

    16. \(f(x) =2x^3+x^2+3\)

    17. \(f(x) =x^3-3x^2+3x-1\)

    18. \(f(x) =\frac{1}{x^2-2x+2}\)

    19. \(f(x) =\frac{x^2-4}{x^2-1}\)

    20. \(f(x) =\frac{x}{x^2-2x-8}\)

    21. \(f(x) =\frac{(x-2)^{2/3}}{x}\)

    22. \(f(x) =\sin x\cos x\text{ on }(-\pi,\pi)\)

    23. \(f(x) = 5x^2-5x\)

    Review

    24. Consider \(f(x)=x^2-3x+5\) on [-1,2]; find \(c\) guaranteed by the Mean Value Theorem.

    25. Consider \(f(x)=\sin x\text{ on }[-\pi/2, \pi/2]\); find \(c) guaranteed by the Mean Value Theorem.

    3.4: Concavity and the Second Derivative

    Terms and Concepts

    1. Sketch a graph of a function \(f(x)\) that is concave up on (0,1) and is concave down on (1,2).

    2. Sketch a graph of a function \(f(x)\) that is:
    (a) Increasing, concave up on (0,1),
    (b) increasing, concave down on (1,2),
    (c) decreasing, concave down on (2,3) and
    (d) increasing, concave down on (3,4).

    3. Is it possible for a function to be increasing and concave down on \((0,\infty)\) with a horizontal asymptote of \(y=1\)? If so, give a sketch of such a function.

    4. Is it possible for a function to be increasing and concave upon \((0,\infty)\) with a horizontal asymptote of \(y=1\)? If so, give a sketch of such a function.

    Problems

    In Exercises 5-15, a function \(f(x)\) is given.
    (a) Compute \(f''(x)\).
    (b) Graph \(f \text{ and }f''\) on the same axes (using technology is permitted) and verify Theorem 31.

    5. \(f(x)=-7x+3\)

    6. \(f(x)=-4x^2+3x-8\)

    7. \(f(x)=4x^2+3x-8\)

    8. \(f(x)=x^3-3x^2+x-1\)

    9. \(f(x)=-x^3+x^2-2x+5\)

    10. \(f(x)=\cos x\)

    11. \(f(x)=\sin x\)

    12. \(f(x) =\tan x\)

    13. \(f(x)=\frac{1}{x^2+1}\)

    14. \(f(x) =\frac{1}{x}\)

    15. \(f(x) = \frac{1}{x^2}\)

    In Exercises 16-28, a function \(f(x)\) is given.
    (a) Find possible points of inflection of \(f\)
    (b) Create a number line to determine the intervals on which \(f\) is concave up or concave down.

    16. \(f(x)=x^2-2x+1\)

    17. \(f(x)=-x^2-5x+7\)

    18. \(f(x)=x^3-x+1\)

    19. \(f(x)=2x^3-3x^2+9x+5\)

    20. \(f(x)=\frac{x^4}{4}+\frac{x^3}{3}-2x+3\)

    21. \(f(x)=-3x^4+8x^3+6x^2-24x+2\)

    22. \(f(x)=x^4-4x^3+6x^2-4x+1\)

    23. \(f(x)=\frac{1}{x^2+1}\)

    24. \(f(x)=\frac{x}{x^2-1}\)

    25. \(f(x)=\sin x+\cos x\text{ on }(-\pi,\pi)\)

    26. \(f(x)=x^2e^x\)

    27. \(f(x)=x^2\ln x\)

    28. \(f(x)=e^{-x^2}\)

    In Exercises 29-41, a function \(f(x)\) is given. Find the critical points of \(f\) and use the Second Derivative Test, when possible, to determine the relative extrema. (Note: these are the same functions as in exercises 16-28.)

    29. \(f(x)=x^2-2x+1\)

    30. \(f(x)=-x^2-5x+7\)

    31. \(f(x)=x^3-x+1\)

    32. \(f(x)=2x^3-3x^2+9x+5\)

    33. \(f(x)=\frac{x^4}{4}+\frac{x^3}{3}-2x+3\)

    34. \(f(x)=-3x^4+8x^3+6x^2-24x+2\)

    35. \(f(x)=x^4-4x^3+6x^2-4x+1\)

    36. \(f(x)=\frac{1}{x^2+1}\)

    37. \(f(x)=\frac{x}{x^2-1}\)

    38. \(f(x)=\sin x+\cos x\text{ on }(-\pi,\pi)\)

    39. \(f(x)=x^2e^x\)

    40. \(f(x)=x^2\ln x\)

    41. \(f(x)=e^{-x^2}\)

    In Exercises 42-54, a function \(f(x)\) is given. Find the x values where \(f'(x)\) has a relative maximum or minimum. (Note: these are the same functions as in Exercises 16-28.)

    42. \(f(x)=x^2-2x+1\)

    43. \(f(x)=-x^2-5x+7\)

    44. \(f(x)=x^3-x+1\)

    45. \(f(x)=2x^3-3x^2+9x+5\)

    46. \(f(x)=\frac{x^4}{4}+\frac{x^3}{3}-2x+3\)

    47. \(f(x)=-3x^4+8x^3+6x^2-24x+2\)

    48. \(f(x)=x^4-4x^3+6x^2-4x+1\)

    49. \(f(x)=\frac{1}{x^2+1}\)

    50. \(f(x)=\frac{x}{x^2-1}\)

    51. \(f(x)=\sin x+\cos x\text{ on }(-\pi,\pi)\)

    52. \(f(x)=x^2e^x\)

    53. \(f(x)=x^2\ln x\)

    54. \(f(x)=e^{-x^2}\)

    3.5: Curve Sketching

    Terms and Concepts

    1. Why is sketching curves by hand beneficial even though technology is ubiquitous?

    2. What does “ubiquitous” mean?

    3. T/F: When sketching graphs of functions, it is useful to find the critical points.

    4. T/F: When sketching graphs of functions, it is useful to find the possible points of inflection.

    5. T/F: When sketching graphs of functions, it is useful to find the horizontal and vertical asymptotes.

    Problems

    In Exercises 6-11, practice using Key Idea 4 by applying the principles to the given functions with familiar graphs.

    6. \(f(x) =2x+4\)

    7. \(f(x) =-x^2+1\)

    8. \(f(x) =\sin x\)

    9. \(f(x) =e^x\)

    10. \(f(x) =\frac{1}{x}\)

    11. \(f(x) =\frac{1}{x^2}\)

    In Exercises 12-25, sketch a graph of the given function using Key Idea 4. Show all work; check your answer with technology.

    12. \(f(x) =x^3-2x^2+4x+1\)

    13. \(f(x) =-x^3+5x^2-3x+2\)

    14. \(f(x) =x^3+3x^2+3x+1\)

    15. \(f(x) =x^3-x^2-x+1\)

    16. \(f(x) =(x-2)\ln (x-2)\)

    17. \(f(x) =(x-2)^2\ln (x-2)\)

    18. \(f(x) =\frac{x^2-4}{x^2}\)

    19. \(f(x) =\frac{x^2-4x+3}{x^2-6x+8}\)

    20. \(f(x) =\frac{x^2-2x+1}{x^2-6x+8}\)

    21. \(f(x) =x\sqrt{x+1}\)

    22. \(f(x) =x^2e^x\)

    23. \(f(x) =\sin x\cos x \text{ on }[-\pi,\pi]\)

    24. \(f(x) =(x-3)^{2/3}+2\)

    25. \(f(x) =\frac{(x-1)^{2/3}}{x}\)

    In Exercises 26-28, a function with the parameters \(a\) and \(b\) are given. Describe the critical points and possible points of inflection of \(f\) in terms of \(a\) and \(b\).

    26. \(f(x) =\frac{a}{x^2+b^2}\)

    27. \(f(x) =\sin (ax+b)\)

    28. \(f(x) = (x-a)(x-b)\)

    29. Given \(x^2+y^2=1\), use implicit differentiation to find \(frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). Use this information to justify the sketch of the unit circle.