# 5.E: Curve Sketching (Exercises)

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- 3457

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These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

## 5.1: Maxima and Minima

In problems 1--12, find all local maximum and minimum points \((x,y)\) by the method of this section.

**Ex 5.1.1** \( y=x^2-x\) (answer)

**Ex 5.1.2** \( y=2+3x-x^3\) (answer)

**Ex 5.1.3** \( y=x^3-9x^2+24x\) (answer)

**Ex 5.1.4** \( y=x^4-2x^2+3\) (answer)

**Ex 5.1.5** \( y=3x^4-4x^3\) (answer)

**Ex 5.1.6** \( y=(x^2-1)/x\) (answer)

**Ex 5.1.7** \( y=3x^2-(1/x^2)\) (answer)

**Ex 5.1.8** \(y=\cos(2x)-x\) (answer)

**Ex 5.1.9** \( f(x) =\cases{ x-1 & \(x < 2\) \cr x^2 & \(x\geq 2$\cr}\) (answer)

**Ex 5.1.10** \exercise \( f(x) =\cases{x-3 & \(x < 3\) \cr x^3 & \(3\leq x \leq 5$\cr 1/x &$x>5$\cr}\) (answer)

**Ex 5.1.11** \( f(x) = x^2 - 98x + 4\) (answer)

**Ex 5.1.12 \(** f(x) =\cases{ -2 & \(x = 0\) \cr 1/x^2 &$x \neq 0$\cr}\) (answer)

**Ex 5.1.13** For any real number \(x\) there is a unique integer \(n\) such that \(n \leq x < n +1\), and the greatest integer function is defined as \(\lfloor x\rfloor = n\). Where are the critical values of the greatest integer function? Which are local maxima and which are local minima?

**Ex 5.1.14 **Explain why the function \(f(x) =1/x\) has no local maxima or minima.

**Ex 5.1.15** How many critical points can a quadratic polynomial function have? (answer)

**Ex 5.1.16** Show that a cubic polynomial can have at most two critical points. Give examples to show that a cubic polynomial can have zero, one, or two critical points.

**Ex 5.1.17** Explore the family of functions \( f(x) = x^3 + cx +1\) where \(c\) is a constant. How many and what types of local extremes are there? Your answer should depend on the value of \(c\), that is, different values of \(c\) will give different answers.

**Ex 5.1.18 **We generalize the preceding two questions. Let \(n\) be a positive integer and let \(f\) be a polynomial of degree \(n\). How many critical points can \(f\) have? (Hint: Recall the **Fundamental Theorem of Algebra**, which says that a polynomial of degree \(n\) has at most \(n\) roots.)

## 5.2: The First Derivative Test

In 1--13, find all critical points and identify them as local maximum points, local minimum points, or neither.

**Ex 5.2.1 **\( y=x^2-x\) (answer)

**Ex 5.2.2** \( y=2+3x-x^3\) (answer)

**Ex 5.2.3** \( y=x^3-9x^2+24x\) (answer)

**Ex 5.2.4** \( y=x^4-2x^2+3\) (answer)

**Ex 5.2.5** \( y=3x^4-4x^3\) (answer)

**Ex 5.2.6** \( y=(x^2-1)/x\) (answer)

**Ex 5.2.7** \( y=3x^2-(1/x^2)\) (answer)

**Ex 5.2.8** \(y=\cos(2x)-x\) (answer)

**Ex 5.2.9** \( f(x) = (5-x)/(x+2)\) (answer)

**Ex 5.2.10 **\( f(x) = |x^2 - 121|\) (answer)

**Ex 5.2.11** \( f(x) = x^3/(x+1)\) (answer)

**Ex 5.2.12** \( f(x)= \cases{x^2 \sin(1/x) & \(x\neq 0\) \cr 0 & \(x=0\)\cr}\)

**Ex 5.2.13** \( f(x) = \sin ^2 x\) (answer)

**Ex 5.2.14 **Find the maxima and minima of \(f(x)=\sec x\). (answer)

**Ex 5.2.15** Let \(\ds f(\theta) = \cos^2(\theta) - 2\sin(\theta)\). Find the intervals where \(f\) is increasing and the intervals where \(f\) is decreasing in \([0,2\pi]\). Use this information to classify the critical points of \(f\) as either local maximums, local minimums, or neither. (answer)

**Ex 5.2.16** Let \(r>0\). Find the local maxima and minima of the function \(\ds f(x) =\sqrt{r^2 -x^2 }\) on its domain \([-r,r]\).

**Ex 5.2.17** Let \(f(x) =a x^2 + bx + c\) with \(a\neq 0\). Show that \(f\) has exactly one critical point using the first derivative test. Give conditions on \(a\) and \(b\) which guarantee that the critical point will be a maximum. It is possible to see this without using calculus at all; explain.

## 5.3: The Second Derivative Test

Find all local maximum and minimum points by the second derivative test.

**Ex 5.3.1**\( y=x^2-x\) (answer)

**Ex 5.3.2**\( y=2+3x-x^3\) (answer)

**Ex 5.3.3**\( y=x^3-9x^2+24x\) (answer)

**Ex 5.3.4**\( y=x^4-2x^2+3\) (answer)

**Ex 5.3.5**\( y=3x^4-4x^3\) (answer)

**Ex 5.3.6**\( y=(x^2-1)/x\) (answer)

**Ex 5.3.7**\( y=3x^2-(1/x^2)\) (answer)

**Ex 5.3.8**\)y=\cos(2x)-x\) (answer)

**Ex 5.3.9**\( y = 4x+\sqrt{1-x}\) (answer)

**Ex 5.3.10**\( y = (x+1)/\sqrt{5x^2 + 35}\) (answer)

**Ex 5.3.11**\( y= x^5 - x\) (answer)

**Ex 5.3.12**\( y = 6x +\sin 3x\) (answer)

**Ex 5.3.13**\( y = x+ 1/x\) (answer)

**Ex 5.3.14**\( y = x^2+ 1/x\) (answer)

**Ex 5.3.15**\( y = (x+5)^{1/4}\) (answer)

**Ex 5.3.16**\( y = \tan^2 x\) (answer)

**Ex 5.3.17**\( y =\cos^2 x - \sin^2 x\) (answer)

**Ex 5.3.18**\( y = \sin^3 x\) (answer)

## Exercises 5.4

Describe the concavity of the functions in 1--18.

**Ex 5.4.1** \( y=x^2-x\) (answer)

**Ex 5.4.2** \( y=2+3x-x^3\) (answer)

**Ex 5.4.3** \( y=x^3-9x^2+24x\) (answer)

**Ex 5.4.4** \( y=x^4-2x^2+3\) (answer)

**Ex 5.4.5** \( y=3x^4-4x^3\) (answer)

**Ex 5.4.6** \( y=(x^2-1)/x\) (answer)

**Ex 5.4.7** \( y=3x^2-(1/x^2)\) (answer)

**Ex 5.4.8**$y=\sin x + \cos x\) (answer)

**Ex 5.4.9** \( y = 4x+\sqrt{1-x}\) (answer)

**Ex 5.4.10** \( y = (x+1)/\sqrt{5x^2 + 35}\) (answer)

**Ex 5.4.11** \( y= x^5 - x\) (answer)

**Ex 5.4.12** \( y = 6x + \sin 3x\) (answer)

**Ex 5.4.13** \( y = x+ 1/x\) (answer)

**Ex 5.4.14** \( y = x^2+ 1/x\) (answer)

**Ex 5.4.15** \( y = (x+5)^{1/4}\) (answer)

**Ex 5.4.16** \( y = \tan^2 x\) (answer)

**Ex 5.4.17** \( y =\cos^2 x - \sin^2 x\) (answer)

**Ex 5.4.18** \( y = \sin^3 x\) (answer)

**Ex 5.4.19**Identify the intervals on which the graph of the function \( f(x) = x^4-4x^3 +10\) is of one of these four shapes: concave up and increasing; concave up and decreasing; concave down and increasing; concave down and decreasing. (answer)

**Ex 5.4.20**Describe the concavity of \( y = x^3 + bx^2 + cx + d\). You will need to consider different cases, depending on the values of the coefficients.

**Ex 5.4.21**Let \(n\) be an integer greater than or equal to two, and suppose \(f\) is a polynomial of degree \(n\). How many inflection points can \(f\) have? Hint: Use the second derivative test and the fundamental theorem of algebra.

## 5.5: Asymptotes and Other Things to Look For

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts.

**Ex 5.5.1** \( y=x^5-5x^4+5x^3\)

**Ex 5.5.2** \( y=x^3-3x^2-9x+5\)

**Ex 5.5.3** \( y=(x-1)^2(x+3)^{2/3}\)

**Ex 5.5.4** \( x^2+x^2y^2=a^2y^2\), \(a>0\).

**Ex 5.5.5** \( y=xe^x\)

**Ex 5.5.6** \( y=(e^x+e^{-x})/2\)

**Ex 5.5.7** \( y=e^{-x}\cos x\)

**Ex 5.5.8** \( y=e^x-\sin x\)

**Ex 5.5.9** \( y=e^x/x\)

**Ex 5.5.10** \( y = 4x+\sqrt{1-x}\)

**Ex 5.5.11** \( y = (x+1)/\sqrt{5x^2 + 35}\)

**Ex 5.5.12** \( y= x^5 - x\)

**Ex 5.5.13** \( y = 6x + \sin 3x\)

**Ex 5.5.14** \( y = x+ 1/x\)

**Ex 5.5.15** \( y = x^2+ 1/x\)

**Ex 5.5.16** \( y = (x+5)^{1/4}\)

**Ex 5.5.17** \( y = \tan^2 x\)

**Ex 5.5.18** \( y =\cos^2 x - \sin^2 x\)

**Ex 5.5.19** \( y = \sin^3 x\)

**Ex 5.5.20** \( y=x(x^2+1)\)

**Ex 5.5.21** \( y=x^3+6x^2 + 9x\)

**Ex 5.5.22** \( y=x/(x^2-9)\)

**Ex 5.5.23** \( y=x^2/(x^2+9)\)

**Ex 5.5.24** \( y=2\sqrt{x} - x\)

**Ex 5.5.25** \( y=3\sin(x) - \sin^3(x)\), for \(x\in[0,2\pi]\)

**Ex 5.5.26** \( y=(x-1)/(x^2)\)

For each of the following five functions, identify any vertical and horizontal asymptotes, and identify intervals on which the function is concave up and increasing; concave up and decreasing; concave down and increasing; concave down and decreasing.

**Ex 5.5.27 \( **f(\theta)=\sec(\theta)\)

**Ex 5.5.28** \( f(x) = 1/(1+x^2)\)

**Ex 5.5.29** \( f(x) = (x-3)/(2x-2)\)

**Ex 5.5.30** \( f(x) = 1/(1-x^2)\)

**Ex 5.5.31** \( f(x) = 1+1/(x^2)\)

**Ex 5.5.32**Let \( f(x) = 1/(x^2-a^2)\), where \(a\geq0\). Find any vertical and horizontal asymptotes and the intervals upon which the given function is concave up and increasing; concave up and decreasing; concave down and increasing; concave down and decreasing. Discuss how the value of \(a\) affects these features.