5.E: Curve Sketching (Exercises)

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)

5.4: Concavity and Inflection Points

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.

Contributors

David Guichard (Whitman College)