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Curve Sketching
Related Topics
Wize University Calculus 1 Textbook > Applications of Differentiation
Curve Sketching
4 Activities
Which of the following statements about the graph of
f
(
x
)
=
e
2
x
−
1
2
x
−
1
f\left(x\right)=\frac{e^{2x-1}}{2x-1}
f
(
x
)
=
2
x
−
1
e
2
x
−
1
is/are true?
(Select all that apply)
The graph has one vertical asymptote
The graph has one horizontal asymptote
lim
x
→
1
2
f
(
x
)
\lim_{x\to\frac{1}{2}}\ f\left(x\right)
lim
x
→
2
1
f
(
x
)
is defined
The function is differentiable everywhere
The function is continuous everywhere
I don't know
Check Submission
More Curve Sketching Questions:
Curve Sketching
Sketch the graph of
f
(
x
)
=
4
x
3
−
3
x
4
f(x) = 4x^3-3x^4
f
(
x
)
=
4
x
3
−
3
x
4
.
Curve Sketching
Sketch out the graph of the function
f
(
x
)
=
x
⋅
e
x
f(x) = x \cdot e^x
f
(
x
)
=
x
⋅
e
x
. The first and second derivatives are given below.
f
′
(
x
)
=
(
x
+
1
)
e
x
,
f
′
′
(
x
)
=
(
x
+
2
)
e
x
f'(x) = (x + 1)e^x \;, \quad f''(x) = (x + 2)e^x
f
′
(
x
)
=
(
x
+
1
)
e
x
,
f
′′
(
x
)
=
(
x
+
2
)
e
x
Curve Sketching
Sketch the graph of
f
(
x
)
=
x
4
−
x
.
f(x)=x\sqrt{4-x}.
f
(
x
)
=
x
4
−
x
.
Curve Sketching
Sketch the graph of
f
(
x
)
=
4
x
3
−
3
x
4
f(x) = 4x^3-3x^4
f
(
x
)
=
4
x
3
−
3
x
4
.
Sketch of Rational Function
Sketch the following function:
f
(
x
)
=
1
x
2
+
3
x
\displaystyle f(x)=\dfrac{1}{x^2+3x}
f
(
x
)
=
x
2
+
3
x
1
.
Sketch of Rational Function
Sketch of the graph of
f
(
x
)
=
x
2
−
9
x
2
+
1
\displaystyle f(x)=\frac{x^2-9}{x^2+1}
f
(
x
)
=
x
2
+
1
x
2
−
9
. The first and the second derivatives are:
f
′
(
x
)
=
20
x
(
x
2
+
1
)
2
and
f
′
′
(
x
)
=
20
[
−
3
x
2
+
1
(
x
2
+
1
)
2
]
{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]}
f
′
(
x
)
=
(
x
2
+
1
)
2
20
x
and
f
′′
(
x
)
=
20
[
(
x
2
+
1
)
2
−
3
x
2
+
1
]
Curve Sketching
Sketch the graph of
f
(
x
)
=
4
x
3
−
3
x
4
f(x) = 4x^3-3x^4
f
(
x
)
=
4
x
3
−
3
x
4
.
Sketch of Rational Function
Sketch the graph of
f
(
x
)
=
x
2
−
9
x
2
+
1
\displaystyle f(x)=\frac{x^2-9}{x^2+1}
f
(
x
)
=
x
2
+
1
x
2
−
9
.
final114
Sketch the graph of
f
(
x
)
=
x
2
3
+
x
2
/
3
f(x)=\dfrac{x^2}{3}+x^{2/3}
f
(
x
)
=
3
x
2
+
x
2/3
Sketch the graph of
f
(
x
)
=
2
x
2
x
2
−
1
f(x) =\dfrac{2x^2}{x^2 − 1}
f
(
x
)
=
x
2
−
1
2
x
2
.
L'Hopital's Rule: Curve Sketching
Sketch the graph of
f
(
x
)
=
x
e
−
x
2
2
f\left(x\right)=xe^{-\frac{x^2}{2}}
f
(
x
)
=
x
e
−
2
x
2
Practice: Sketch of Piecewise Function
Q.
\textbf{Q.}
Q.
Sketch the graph of
f
(
x
)
=
{
2
e
x
if
x
<
0
x
2
+
6
3
(
x
+
1
)
if
x
≥
0
{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.}
f
(
x
)
=
⎩
⎨
⎧
2
e
x
if
x
<
0
3
(
x
+
1
)
x
2
+
6
if
x
≥
0
Sketch of Rational Function
Sketch the following function:
f
(
x
)
=
1
x
2
+
3
x
\displaystyle f(x)=\dfrac{1}{x^2+3x}
f
(
x
)
=
x
2
+
3
x
1
.
Sketch of Rational Function
Sketch of the graph of
f
(
x
)
=
x
2
−
9
x
2
+
1
\displaystyle f(x)=\frac{x^2-9}{x^2+1}
f
(
x
)
=
x
2
+
1
x
2
−
9
. The first and the second derivatives are:
f
′
(
x
)
=
20
x
(
x
2
+
1
)
2
and
f
′
′
(
x
)
=
20
[
−
3
x
2
+
1
(
x
2
+
1
)
2
]
{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]}
f
′
(
x
)
=
(
x
2
+
1
)
2
20
x
and
f
′′
(
x
)
=
20
[
(
x
2
+
1
)
2
−
3
x
2
+
1
]
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
)
A(t)
A
(
t
)
be the yellow graph,
B
(
t
)
B(t)
B
(
t
)
be the blue graph and
C
(
t
)
C(t)
C
(
t
)
be the red graph.
Curve Sketching
Let
y
=
f
(
x
)
=
(
x
−
2
)
2
(
x
+
2
)
y = f(x) = (x - 2)^2(x + 2)
y
=
f
(
x
)
=
(
x
−
2
)
2
(
x
+
2
)
a) Does
f
(
x
)
f(x)
f
(
x
)
have any horizontal asymptotes?
Practice: Curve Sketching
Q.
\textbf{Q.}
Q.
Sketch of the graph of
f
(
x
)
=
x
2
−
9
x
2
+
1
\displaystyle f(x)=\frac{x^2-9}{x^2+1}
f
(
x
)
=
x
2
+
1
x
2
−
9
. The first and the second derivatives are
f
′
(
x
)
=
20
x
(
x
2
+
1
)
2
and
f
′
′
(
x
)
=
20
[
−
3
x
2
+
1
(
x
2
+
1
)
2
]
\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]}
f
′
(
x
)
=
(
x
2
+
1
)
2
20
x
and
f
′′
(
x
)
=
20
[
(
x
2
+
1
)
2
−
3
x
2
+
1
]
The graph of the velocity,
v
(
t
)
v\left(t\right)
v
(
t
)
, of a particle is depicted below. Let
f
(
t
)
f\left(t\right)
f
(
t
)
denote the position of the particle at time
t
t
t
Curve Sketching
Graph
y
=
x
e
−
3
x
y=xe^{-3x}
y
=
x
e
−
3
x
for the domain
[
0
,
1
]
[0,1]
[
0
,
1
]
. Label any critical point
or inflection point.
Sketch the graph of
f
(
x
)
=
x
4
+
x
2
f(x) =\dfrac{x}{4 + x^2}
f
(
x
)
=
4
+
x
2
x
.
Curve Sketching
Sketch the graph of
f
(
x
)
=
4
x
3
−
3
x
4
f(x) = 4x^3-3x^4
f
(
x
)
=
4
x
3
−
3
x
4
.
Sketch the graph of
f
(
x
)
=
2
x
2
x
2
−
1
f(x) =\dfrac{2x^2}{x^2 − 1}
f
(
x
)
=
x
2
−
1
2
x
2
.
Sketch the graph of
f
(
x
)
=
x
2
3
+
x
2
/
3
f(x)=\dfrac{x^2}{3}+x^{2/3}
f
(
x
)
=
3
x
2
+
x
2/3
Sketch the graph of
f
(
x
)
=
x
2
+
9
x
−
3
f(x) =\dfrac{x^2 + 9}{x − 3}
f
(
x
)
=
x
−
3
x
2
+
9
.
Sketch the graph of
f
(
x
)
=
2
x
2
x
2
−
1
f(x) =\dfrac{2x^2}{x^2 − 1}
f
(
x
)
=
x
2
−
1
2
x
2
.
Sketch of Rational Function
Sketch the following function:
f
(
x
)
=
1
x
2
+
3
x
\displaystyle f(x)=\dfrac{1}{x^2+3x}
f
(
x
)
=
x
2
+
3
x
1
.
Sketch of Rational Function
Sketch of the graph of
f
(
x
)
=
x
2
−
9
x
2
+
1
\displaystyle f(x)=\frac{x^2-9}{x^2+1}
f
(
x
)
=
x
2
+
1
x
2
−
9
. The first and the second derivatives are:
f
′
(
x
)
=
20
x
(
x
2
+
1
)
2
and
f
′
′
(
x
)
=
20
[
−
3
x
2
+
1
(
x
2
+
1
)
2
]
{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]}
f
′
(
x
)
=
(
x
2
+
1
)
2
20
x
and
f
′′
(
x
)
=
20
[
(
x
2
+
1
)
2
−
3
x
2
+
1
]
Practice: Sketch of Piecewise Function
Q.
\textbf{Q.}
Q.
Sketch the graph of
f
(
x
)
=
{
2
e
x
if
x
<
0
x
2
+
6
3
(
x
+
1
)
if
x
≥
0
{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.}
f
(
x
)
=
⎩
⎨
⎧
2
e
x
if
x
<
0
3
(
x
+
1
)
x
2
+
6
if
x
≥
0
final114
Sketch the graph of
f
(
x
)
=
x
2
3
+
x
2
/
3
f(x)=\dfrac{x^2}{3}+x^{2/3}
f
(
x
)
=
3
x
2
+
x
2/3
Sketch the graph of
f
(
x
)
=
2
x
2
x
2
−
1
f(x) =\dfrac{2x^2}{x^2 − 1}
f
(
x
)
=
x
2
−
1
2
x
2
.
L'Hopital's Rule: Curve Sketching
Sketch the graph of
f
(
x
)
=
x
e
−
x
2
2
f\left(x\right)=xe^{-\frac{x^2}{2}}
f
(
x
)
=
x
e
−
2
x
2
Sketch the graph of
f
(
x
)
=
x
2
3
+
x
2
/
3
f(x)=\frac{x^2}{3}+x^{2/3}
f
(
x
)
=
3
x
2
+
x
2/3
.
Curve Sketching
Sketch the graph of
f
(
x
)
=
x
4
−
x
.
f(x)=x\sqrt{4-x}.
f
(
x
)
=
x
4
−
x
.
Curve Sketching
Sketch the graph of
f
(
x
)
=
x
e
−
x
2
/
2
.
f\left(x\right)=xe^{-x^2/2}.
f
(
x
)
=
x
e
−
x
2
/2
.
For the graph of
f
(
x
)
=
2
x
2
x
2
−
1
f(x) =\dfrac{2x^2}{x^2 − 1}
f
(
x
)
=
x
2
−
1
2
x
2
,
Find the intervals of increase, decrease, concave up, and concave down.
Determine minima and maxima.
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
)
A(t)
A
(
t
)
be the yellow graph,
B
(
t
)
B(t)
B
(
t
)
be the blue graph and
C
(
t
)
C(t)
C
(
t
)
be the red graph.
Curve Sketching
Which of the following graphs is a sketch of the function
f
(
x
)
=
x
+
e
−
x
f(x) = x + e^{-x}
f
(
x
)
=
x
+
e
−
x
for all
x
x
x
.
a)
b)
Curve Sketching
Let
f
(
x
)
f(x)
f
(
x
)
be a smooth function (it is infinitely differentiable). on the interval
[
−
5
,
4
]
[-5, 4]
[
−
5
,
4
]
. The following table lists the relevant information necessary to create a sketch of the function on the interval
[
−
5
,
4
]
[-5, 4]
[
−
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
)
f(x), f'(x)
f
(
x
)
,
f
′
(
x
)
or
f
′
′
(
x
)
f''(x)
f
′′
(
x
)
at a particular point, or over a region.
Curve Sketching
Sketch out the graph of the function
f
(
x
)
=
x
⋅
e
x
f(x) = x \cdot e^x
f
(
x
)
=
x
⋅
e
x
. The first and second derivatives are given below.
f
′
(
x
)
=
(
x
+
1
)
e
x
,
f
′
′
(
x
)
=
(
x
+
2
)
e
x
f'(x) = (x + 1)e^x \;, \quad f''(x) = (x + 2)e^x
f
′
(
x
)
=
(
x
+
1
)
e
x
,
f
′′
(
x
)
=
(
x
+
2
)
e
x
Curve Sketching
Find all zeroes, minima, maxima and inflection points of
f
(
x
)
=
x
5
−
x
3
f(x) = x^5 - x^3
f
(
x
)
=
x
5
−
x
3
. Sketch the graph of the function.
Practice
Sketch of the graph of
f
(
x
)
=
1
x
2
+
1
\displaystyle f(x)=\frac{1}{x^2+1}
f
(
x
)
=
x
2
+
1
1
. The first and the second derivatives are
f
′
(
x
)
=
−
2
x
(
x
2
+
1
)
2
and
f
′
′
(
x
)
=
6
x
2
−
2
(
x
2
+
1
)
3
.
\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}.}
f
′
(
x
)
=
(
x
2
+
1
)
2
−
2
x
and
f
′′
(
x
)
=
(
x
2
+
1
)
3
6
x
2
−
2
.
f
(
x
)
=
x
e
−
3
x
f\left(x\right)=xe^{-3x}
f
(
x
)
=
x
e
−
3
x
