¹⁰⁵ views
ubuntu2204

Kernel: SageMath 10.0

Function Analysis Part 1 Assignment

Analyze the following functions using the steps from class.

Some steps have been done for you - make sure you run the function definitions again.

Question 1

[5 points] $\quad\displaystyle f(x)=\frac{9x-1}{x^2+110}$

In [11]:

f(x)=(9*x-1)/(x^2+110)
show(f(x))

Out[11]:

$\displaystyle \frac{9 \, x - 1}{x^{2} + 110}$

Step 1: Find the domain of $f$ .

The domain is $\R$ .

Step 2: Find the derivative $f'$ .

In [12]:

df(x)=derivative(f(x),x)
show(df(x))

Out[12]:

$\displaystyle -\frac{2 \, {\left(9 \, x - 1\right)} x}{{\left(x^{2} + 110\right)}^{2}} + \frac{9}{x^{2} + 110}$

Step 3: Find the critical points of $f$ (where $f'$ is $0$ or undefined).

In [13]:

solve(df(x)==0,x)

Out[13]:

[x == -1/9*sqrt(8911) + 1/9, x == 1/9*sqrt(8911) + 1/9]

You should convert the solutions above to decimal.

In [14]:

N(-1/9*sqrt(8911)+1/9)

Out[14]:

-10.3775659112312

In [15]:

N(1/9*sqrt(8911)+1/9)

Out[15]:

10.5997881334535

Step 4: See if the sign of $f'$ actually changes at the critical points of $f$ , and determine whether $f$ has a local maximum or local minimum at these points.

In [16]:

df(-10.5)

Out[16]:

-0.000479282004635637

In [17]:

df(-10.3)

Out[17]:

0.000312453902207771

In [18]:

df(10.45)

Out[18]:

0.000584342440431880

In [19]:

df(10.7)

Out[19]:

-0.000377214272833236

Step 5: Find the second derivative $f''$ .

In [20]:

d2f(x)=derivative(f(x),x,2)
show(d2f(x))

Out[20]:

$\displaystyle \frac{8 \, {\left(9 \, x - 1\right)} x^{2}}{{\left(x^{2} + 110\right)}^{3}} - \frac{2 \, {\left(9 \, x - 1\right)}}{{\left(x^{2} + 110\right)}^{2}} - \frac{36 \, x}{{\left(x^{2} + 110\right)}^{2}}$

Step 6: Find the critical points of $f'$ (where $f''$ is $0$ or undefined).

In [21]:

solve(d2f(x)==0,x)

Out[21]:

[x == -1/2*(8911/81*I*sqrt(110) + 8911/729)^(1/3)*(I*sqrt(3) + 1) - 8911/162*(-I*sqrt(3) + 1)/(8911/81*I*sqrt(110) + 8911/729)^(1/3) + 1/9, x == -1/2*(8911/81*I*sqrt(110) + 8911/729)^(1/3)*(-I*sqrt(3) + 1) - 8911/162*(I*sqrt(3) + 1)/(8911/81*I*sqrt(110) + 8911/729)^(1/3) + 1/9, x == (8911/81*I*sqrt(110) + 8911/729)^(1/3) + 8911/81/(8911/81*I*sqrt(110) + 8911/729)^(1/3) + 1/9]

You should convert the solutions above to decimal.

In [5]:

N(-1/2*(8911/81*I*sqrt(110) + 8911/729)^(1/3)*(I*sqrt(3) +1)-8911/162*(-I*sqrt(3) +1)/(8911/81*I*sqrt(110) + 8911/729)^(1/3)+ 1/9)

Out[5]:

0.0370358054710737 + 5.32907051820075e-15*I

In [22]:

N(-1/2*(8911/81*I*sqrt(110) + 8911/729)^(1/3)*(I*sqrt(3) +1)-8911/162*(-I*sqrt(3) +1)/(8911/81*I*sqrt(110) + 8911/729)^(1/3)+ 1/9)

Out[22]:

0.0370358054710737 + 5.32907051820075e-15*I

In [23]:

N(-1/2*(8911/81*I*sqrt(110) + 8911/729)^(1/3)*(I*sqrt(3) +1)-8911/162*(-I*sqrt(3) +1)/(8911/81*I*sqrt(110) + 8911/729)^(1/3)+ 1/9)

Out[23]:

0.0370358054710737 + 5.32907051820075e-15*I

In [7]:

N(1/6*sqrt(39)+1)

Out[7]:

2.04083299973307

In [8]:

N(-1/6*sqrt(39)+1)

Out[8]:

-0.0408329997330663

Step 7: See if the sign of $f''$ actually changes at the critical points of $f'$ , and determine whether $f$ has an inflection point at these points.

In [25]:

d2f(0)
N(1/6050)

Out[25]:

0.000165289256198347

In [26]:

d2f(0.1)

Out[26]:

-0.000280946661976095

In [27]:

d2f(-18.2)

Out[27]:

-0.0000253027535196296

In [28]:

d2f(-17.2)

Out[28]:

0.000134953803168578

In [29]:

d2f(18.1)

Out[29]:

-0.0000301219469454848

In [30]:

d2f(18.4)

Out[30]:

0.0000113426926653792

In [0]:

#inflection points at x=-18.02, 0.037, 18.31

Step 8: Find the $x$ - and $y$ -intercepts.

In [31]:

solve(f(x)==0,x)

Out[31]:

[x == (1/9)]

In [32]:

f(0)

Out[32]:

-1/110

Step 9: Determine the end behavior.

In [33]:

limit(f(x),x=infinity)

Out[33]:

0

In [34]:

limit(f(x),x=-infinity)

Out[34]:

0

Step 10: Make an informed graph. Mark any $x$ - and $y$ -intercepts, relative maxima and minima, and inflection points.

In [35]:

f(-10.38)

Out[35]:

-0.433627684569615

In [0]:

f(10.6)

In [0]:

f(0.037)

In [0]:

f(-18.02)

In [0]:

f(18.31)

In [36]:

plot (f(x), xmin=-100, xmax=100, ymin=-0.5, ymax=0.5)+point([(-10.38,-0.43), (10.60,0.42), (0.037, -0.006), (-18.02, -0.375), (18.31, 0.368), (1/9,0),(0, -1/110)], color='black', size=20)

Out[36]:

Step 11: Discuss absolute max/min, increasing/decreasing, concave up/down.

In [37]:

#Absolute Max=0.42, at x=10.60
#Absolute Min=-0.43, at x=10.38
#inc (-10.38, 10.60)
#dec (-infinity, -10.38) (10.60, infinity)
#cu (-18.02, 0.037), (18.31, infinity)
#cd (-infinity, -18.02), (0.037, 18.31)

Question 2

[5 points] $\quad g(x)=2x^4-8x^3-x^2+30x$

In [38]:

g(x)=2*x^4-8*x^3-x^2+30*x
show(g(x))

Out[38]:

$\displaystyle 2 \, x^{4} - 8 \, x^{3} - x^{2} + 30 \, x$

Step 1: Find the domain of $g$ .

The domain is $\R$ .

Step 2: Find the derivative $g'$ .

In [39]:

dg(x)=derivative(g(x),x)
show(dg(x))

Out[39]:

$\displaystyle 8 \, x^{3} - 24 \, x^{2} - 2 \, x + 30$

Step 3: Find the critical points of $g$ (where $g'$ is $0$ or undefined).

In [40]:

solve(dg(x)==0,x)

Out[40]:

[x == (5/2), x == -1, x == (3/2)]

Step 4: See if the sign of $g'$ actually changes at the critical points of $g$ , and determine whether $g$ has a local maximum or local minimum at these points.

In [41]:

dg(2)

Out[41]:

-6

In [42]:

dg(3)

Out[42]:

24

In [0]:

dg(1)

In [43]:

dg(2)

Out[43]:

-6

In [0]:

dg(-1.5)

In [44]:

dg(-0.5)

Out[44]:

24.0000000000000

Step 5: Find the second derivative $g''$ .

In [45]:

d2g(x)=derivative(g(x),x,2)
show(d2g(x))

Out[45]:

$\displaystyle 24 \, x^{2} - 48 \, x - 2$

Step 6: Find the critical points of $g'$ (where $g''$ is $0$ or undefined).

In [46]:

solve(d2g(x)==0,x)

Out[46]:

[x == -1/6*sqrt(39) + 1, x == 1/6*sqrt(39) + 1]

In [47]:

N(-1/6*sqrt(39)+1)

Out[47]:

-0.0408329997330663

In [48]:

N(1/6 *sqrt(39)+1)

Out[48]:

2.04083299973307

In [0]:

Step 7: See if the sign of $g''$ actually changes at the critical points of $g'$ , and determine whether $g$ has an inflection point at these points.

In [49]:

d2g(-0.05)

Out[49]:

0.460000000000000

In [51]:

d2g(-0.03)

Out[51]:

-0.538400000000000

In [52]:

d2g(2)

Out[52]:

-2

In [53]:

d2g(3)

Out[53]:

70

In [55]:

#inflection point at x=2.041

Step 8: Find the $x$ - and $y$ -intercepts.

[Caution: $g$ has two x-intercepts. When you solve $g(x)=0$ , CoCalc will give you four answers, but only two are real. Convert to decimal, and watch out for scientific notation.]

In [56]:

solve(g(x)==0,x)

Out[56]:

[x == -1/6*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3)*(I*sqrt(3) + 1) - 5/3*(7/4)^(2/3)*(-I*sqrt(3) + 1)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3, x == -1/6*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3)*(-I*sqrt(3) + 1) - 5/3*(7/4)^(2/3)*(I*sqrt(3) + 1)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3, x == 1/3*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 10/3*(7/4)^(2/3)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3, x == 0]

In [14]:

g(0)

Out[14]:

0

In [58]:

N(-1/6*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3)*(I*sqrt(3) + 1) - 5/3*(7/4)^(2/3)*(-I*sqrt(3) + 1)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3)

Out[58]:

2.83551650478519 + 0.967641208530388*I

In [59]:

N( -1/6*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3)*(-I*sqrt(3) + 1) - 5/3*(7/4)^(2/3)*(I*sqrt(3) + 1)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3)

Out[59]:

-1.67103300957038 - 2.74364199474365e-17*I

In [61]:

N(1/3*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 10/3*(7/4)^(2/3)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3)

Out[61]:

2.83551650478519 - 0.967641208530389*I