
# 5.E: Curve Sketching (Exercises)

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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 5.3.9$$y = 4x+\sqrt{1-x}$$ (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 5.4.19Identify 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 5.4.20Describe 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 5.4.21Let $$n$$ 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 5.5.32Let $$f(x) = 1/(x^2-a^2)$$, 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.