Wize University Calculus 1 Textbook > Applications of Differentiation

Curve Sketching

Curve Sketching

Combining our knowledge of the first and second derivatives lets us sketch the graph of functions by hand. We will put together the techniques of finding asymptotes, intervals of increase/decrease, extrema, intervals of concavity, and inflection points.

Procedure for completely Sketching a Curve

Find the domain of $f(x)$ .
Find the $y$ -intercepts and (if possible) the $x$ -intercepts.
Find all the vertical, horizontal and slant asymptotes. The line $y = mx + b$ is a slant asymptote of the graph of $f(x)$ if either $\lim_{x\rightarrow -\infty}\bigg[f(x)-(mx+b)\bigg]=0 \ \text{ or } \ \lim_{x\rightarrow +\infty}\bigg[f(x)-(mx+b)\bigg]=0$ (You might have to do long division)
Compute $f'(x)$ . Use it to find all the critical points and singular points of $f(x)$ . Use this information to find the intervals of increase and decrease and extrema.
Compute $f''(x)$ . Find the points for which $f''(x) = 0$ or $f''(x)$ does not exist. Use this information to find the intervals of concavity and inflection points.
Plot all relevant points, asymptotes, and use the information gathered to plot the graph of $f(x)$ .

Example: Curve Sketching

Sketch the function

$\displaystyle f\left(x\right)=\frac{x^3}{3}-x$

1) Find the Domain

$f(x)$ is a polynomial so the domain is $(-\infin,\infin)$

2) Find the intercepts

x intercept: $\displaystyle\implies f(x)=0=\frac{x^3}{3}-x=x(\frac{x^2}{3}-1) \implies x=0 \text{ and } \frac{x^2}{3}-1=0 \implies x=0,-\sqrt{3},\sqrt{3}$

y intercept: $\displaystyle \implies f(0)=\frac{0^3}{3}-0=0\implies y=0$

3) Find the Asymptotes

$f(x)$ is a polynomial and has no asymptotes.

4) Critical points and intervals of Increasing/decreasing

$f'(x)=x^2-1$

$f'(x)=x^2-1=0 \implies (x-1)(x+1)=0 \implies x=-1,1$

$f'(x)>0 \text{ if }x\text{ is in }(-\infin,-1) \text{ or } (1,\infin)$

$f'(x)<0 \text{ if } x\text{ is in }(-1,1)$

5) Inflection points and intervals of Concavity

$f''(x)=2x$

$f''(x)=2x=0 \implies x=0$

$f''(x)>0 \text{ if } x>0$

$f''(x)<0 \text{ if } x<0$

6) Plot points, asymptotes, and Graph

Mark Yourself Question

Grab a piece of paper and try this problem yourself.
When you're done, check the "I have answered this question" box below.
View the solution and report whether you got it right or wrong.

Sketch of the graph of $\displaystyle f(x)=\frac{1}{x^2+1}$ . The first and the second derivatives are

\boxed{f^{\prime}(x)=\frac{-2x}{\left(x^{2}+1\right)^{2}}\quad\text{ and }\quad f^{\prime\prime}(x)=\frac{6x^{2}-2}{\left(x^{2}+1\right)^3}}

I have answered this question

Practice: Curve Sketching From Derivatives

The first derivative $f'\left(x\right)$ is depicted below

Mark Yourself Question

Grab a piece of paper and try this problem yourself.
When you're done, check the "I have answered this question" box below.
View the solution and report whether you got it right or wrong.

List the critical points of $f\left(x\right)$ and classify them as a max, min, or neither.

I have answered this question

Extra Practice

$f\left(x\right)=xe^{-3x}$

Practice

Sketch of the graph of $\displaystyle f(x)=\frac{1}{x^2+1}$ . The first and the second derivatives are

\boxed{f^{\prime}(x)=\frac{-2x}{\left(x^{2}+1\right)^{2}}\quad\text{ and }\quad f^{\prime\prime}(x)=\frac{6x^{2}-2}{\left(x^{2}+1\right)^3}.}

Curve Sketching

Find all zeroes, minima, maxima and inflection points of $f(x) = x^5 - x^3$ . Sketch the graph of the function.

Curve Sketching

Sketch out the graph of the function $f(x) = x \cdot e^x$ . The first and second derivatives are given below.

f'(x) = (x + 1)e^x \;, \quad f''(x) = (x + 2)e^x

Curve Sketching

Let $f(x)$ be a smooth function (it is infinitely differentiable). on the interval $[-5, 4]$ . The following table lists the relevant information necessary to create a sketch of the function on the interval $[-5, 4]$ . Use this information to graph the function, while labeling zeroes, local minima, local maxima, and inflection points on the function.

The columns of this spreadsheet indicate either the actual value or the sign of $f(x), f'(x)$ or $f''(x)$ at a particular point, or over a region.

Curve Sketching

Which of the following graphs is a sketch of the function $f(x) = x + e^{-x}$ for all $x$ .

a) Illustration for practice question: Curve Sketching

b) Illustration for practice question: Curve Sketching

Curve Sketching

The graphs below represent the position, velocity and acceleration of a mass attached to an ideal spring on a floor with no friction. Identify the correct relationships between these functions.

Let $A(t)$ be the yellow graph, $B(t)$ be the blue graph and $C(t)$ be the red graph.

Curve Sketching

Let

y = f(x) = (x - 2)^2(x + 2)

a) Does $f(x)$ have any horizontal asymptotes?

Curve Sketching

Which of the following statements about the graph of $f\left(x\right)=\frac{e^{2x-1}}{2x-1}$ is/are true?

(Select all that apply)

For the graph of $f(x) =\dfrac{2x^2}{x^2 − 1}$ ,

Find the intervals of increase, decrease, concave up, and concave down.

Determine minima and maxima.

Curve Sketching

Sketch the graph of $f\left(x\right)=xe^{-x^2/2}.$

Curve Sketching

Sketch the graph of $f(x)=x\sqrt{4-x}.$

Sketch the graph of $f(x)=\frac{x^2}{3}+x^{2/3}$ .

L'Hopital's Rule: Curve Sketching

Sketch the graph of $f\left(x\right)=xe^{-\frac{x^2}{2}}$

Sketch the graph of $f(x) =\dfrac{2x^2}{x^2 − 1}$ .

final114

Sketch the graph of $f(x)=\dfrac{x^2}{3}+x^{2/3}$

Practice: Sketch of Piecewise Function

$\textbf{Q.}$ Sketch the graph of

{f(x)=\left\{\begin{array}{ll} 2e^x\quad\qquad\,\,\text{if}\,x<0 \\ \displaystyle\frac{x^2+6}{3(x+1)}\quad\text{if}\,x\geq 0 \end{array}\right.}

Sketch of Rational Function

Sketch of the graph of $\displaystyle f(x)=\frac{x^2-9}{x^2+1}$ . The first and the second derivatives are:

{f^{\prime}(x)=\frac{20x}{(x^2+1)^2}\quad\text{ and }\quad f^{\prime\prime}(x)=20\bigg[\frac{-3x^2+1}{(x^2+1)^2}\bigg]}

Sketch of Rational Function

Sketch the following function: $\displaystyle f(x)=\dfrac{1}{x^2+3x}$ .

Sketch the graph of $f(x) =\dfrac{2x^2}{x^2 − 1}$ .

Sketch the graph of $f(x) =\dfrac{x^2 + 9}{x − 3}$ .

Sketch the graph of $f(x)=\dfrac{x^2}{3}+x^{2/3}$

Sketch the graph of $f(x) =\dfrac{2x^2}{x^2 − 1}$ .

Curve Sketching

Sketch the graph of $f(x) = 4x^3-3x^4$ .

Sketch the graph of $f(x) =\dfrac{x}{4 + x^2}$ .

Curve Sketching

Graph $y=xe^{-3x}$ for the domain $[0,1]$ . Label any critical point

or inflection point.

The graph of the velocity, $v\left(t\right)$ , of a particle is depicted below. Let $f\left(t\right)$ denote the position of the particle at time $t$

Practice: Curve Sketching

$\textbf{Q.}$ Sketch of the graph of $\displaystyle f(x)=\frac{x^2-9}{x^2+1}$ . The first and the second derivatives are

\boxed{f^{\prime}(x)=\frac{20x}{(x^2+1)^2}\quad\text{ and }\quad f^{\prime\prime}(x)=20\bigg[\frac{-3x^2+1}{(x^2+1)^2}\bigg]}

Curve Sketching

The graphs below represent the position, velocity and acceleration of a mass attached to an ideal spring on a floor with no friction. Identify the correct relationships between these functions.

Let $A(t)$ be the yellow graph, $B(t)$ be the blue graph and $C(t)$ be the red graph.

Sketch of Rational Function

Sketch of the graph of $\displaystyle f(x)=\frac{x^2-9}{x^2+1}$ . The first and the second derivatives are:

{f^{\prime}(x)=\frac{20x}{(x^2+1)^2}\quad\text{ and }\quad f^{\prime\prime}(x)=20\bigg[\frac{-3x^2+1}{(x^2+1)^2}\bigg]}

Sketch of Rational Function

Sketch the following function: $\displaystyle f(x)=\dfrac{1}{x^2+3x}$ .

Practice: Sketch of Piecewise Function

$\textbf{Q.}$ Sketch the graph of

{f(x)=\left\{\begin{array}{ll} 2e^x\quad\qquad\,\,\text{if}\,x<0 \\ \displaystyle\frac{x^2+6}{3(x+1)}\quad\text{if}\,x\geq 0 \end{array}\right.}

L'Hopital's Rule: Curve Sketching

Sketch the graph of $f\left(x\right)=xe^{-\frac{x^2}{2}}$

Sketch the graph of $f(x) =\dfrac{2x^2}{x^2 − 1}$ .

final114

Sketch the graph of $f(x)=\dfrac{x^2}{3}+x^{2/3}$

Sketch of Rational Function

Sketch the graph of $\displaystyle f(x)=\frac{x^2-9}{x^2+1}$ .

Curve Sketching

Sketch the graph of $f(x) = 4x^3-3x^4$ .

Sketch of Rational Function

Sketch of the graph of $\displaystyle f(x)=\frac{x^2-9}{x^2+1}$ . The first and the second derivatives are:

{f^{\prime}(x)=\frac{20x}{(x^2+1)^2}\quad\text{ and }\quad f^{\prime\prime}(x)=20\bigg[\frac{-3x^2+1}{(x^2+1)^2}\bigg]}

Sketch of Rational Function

Sketch the following function: $\displaystyle f(x)=\dfrac{1}{x^2+3x}$ .

Curve Sketching

Sketch the graph of $f(x) = 4x^3-3x^4$ .

Curve Sketching

Sketch the graph of $f(x)=x\sqrt{4-x}.$

Curve Sketching

Sketch out the graph of the function $f(x) = x \cdot e^x$ . The first and second derivatives are given below.

f'(x) = (x + 1)e^x \;, \quad f''(x) = (x + 2)e^x

Curve Sketching

Sketch the graph of $f(x) = 4x^3-3x^4$ .

Curve Sketching

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Curve Sketching

Procedure for completely Sketching a Curve

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Example: Curve Sketching

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Practice: Curve Sketching From Derivatives

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Extra Practice