5.E: Curve Sketching (Exercises)
- Page ID
- 3090
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.