a.) Use a graph of $f(x) = x^3(x-2)^4$ to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection.
b.) Use a graph of $f''$ to give better estimates.
a.)
Based from the graph, the function has an upward concavity at intervals, $0 < x < 0.4$ and $ x > 1.3$. On the other hand, the funciton has a downward concavity at interval $x < 0 $ and $0.4 < x < 1.3$. Also the coordinates of the points of inflections can be approximated as $(0,0),(0.4,0.5)$ and $(1.3,0.5)$
$
\begin{equation}
\begin{aligned}
\text{if } f (x) &= x^3(x-2)^4, \text{ then}\\
\\
f'(x) &= x^3 \left[ 4 (x-2)^3 \right] + 3x^2(x-2)^4 && \Longleftarrow \text{(By using Chain and Product Rule)}\\
\\
f''(x) &= x^3 \left[ 12(x - 2)^2 \right] + 3x^2 \left( 4(x-2)^3\right) + 3x^2 \left[4 (x-2)^3 \right]+ 6x (x-2)^4\\
\\
f''(x) &= 12x^3 (x-2)^2 + 12x^2 (x-2)^3 + 12x^2 (x-2)^3 + 6x(x-2)^4\\
\\
f''(x) &= 12x^3(x-2)^2 + 24 x^2 (x-2)^3 + 6x (x-2)^4
\end{aligned}
\end{equation}
$
Based from the graph of $f''$, the function has an upward concavity (where $f''$ is positive) at intervals $0 < x < 0.46$ and $x > 1.28$ On the other hand, the function has downward concavity (where $f''$ is negative) at intervals $x < 0 $ and $ 0.46 < x < 1.28$
Also using the graph of $f''$, the points of inflections (where the slope is 0) are approximately $x = 0.18, \quad x = 0.83, \quad x = 1.57, \quad x = 2$
Wednesday, May 25, 2016
Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 46
Subscribe to:
Post Comments (Atom)
Why is the fact that the Americans are helping the Russians important?
In the late author Tom Clancy’s first novel, The Hunt for Red October, the assistance rendered to the Russians by the United States is impor...
-
There are a plethora of rules that Jonas and the other citizens must follow. Again, page numbers will vary given the edition of the book tha...
-
The poem contrasts the nighttime, imaginative world of a child with his daytime, prosaic world. In the first stanza, the child, on going to ...
-
The given two points of the exponential function are (2,24) and (3,144). To determine the exponential function y=ab^x plug-in the given x an...
-
The play Duchess of Malfi is named after the character and real life historical tragic figure of Duchess of Malfi who was the regent of the ...
-
The only example of simile in "The Lottery"—and a particularly weak one at that—is when Mrs. Hutchinson taps Mrs. Delacroix on the...
-
Hello! This expression is already a sum of two numbers, sin(32) and sin(54). Probably you want or express it as a product, or as an expressi...
-
Macbeth is reflecting on the Weird Sisters' prophecy and its astonishing accuracy. The witches were totally correct in predicting that M...
No comments:
Post a Comment