Question:
Match the given functions with their graphs and the area under the curve over $[-1,1]$:
Function
Graph
Area under curve $[-1,1]$
(i) $f(x) = x$
(b)
$\int_{-1}^{1} x,dx = 0$
(ii) $f(x) =
x
$
(iii) $f(x) = x^2$
(a)
$\int_{-1}^{1} x^2,dx = \frac{2}{3}$
Solution:
(i) $\rightarrow$ (b) $\rightarrow$ (3) (Note: The “3” here is likely a typo or mislabel; area is 0, not “3”. The correct area solution is as above.)
Corrected: (i) $\rightarrow$ (b) $\rightarrow$ area $= 0$
**(ii) $\rightarrow$ (c) $\rightarrow$ area $= 1$
**(iii) $\rightarrow$ (a) $\rightarrow$ area $= \frac{2}{3}$
2. Curves Enclosing Negative Area
Question:
Which of the following curves enclose a negative area on the x-axis in the interval $1$?
Area above the x-axis is positive, below is negative. If the area below the x-axis is more than above, the net area is negative.
Solution:
Curve 2 and Curve 4 enclose a negative area.
3. Cylinder Inscribed in a Circle
Question:
A cylinder of radius $x$ and height $2h$ is inscribed in a circle of radius $R$. Find the values of $h$ for maximum volume and maximum surface area.
Solution:
Volume:
From right triangle: $h^2 + x^2 = R^2$
Volume $V = 2\pi x^2 h = 2\pi (R^2 - h^2) h = 2\pi R^2 h - 2\pi h^3$
$\frac{d^2S}{dh^2}\left(\frac{R}{\sqrt{2}}\right) = -20\pi < 0$ $\Rightarrow$ maximum at $h = \frac{R}{\sqrt{2}}$
4. Minimum Difference of Functions
Question:
Given $f_2(x) = x$ and $g_2(x) = e^x$ are increasing functions, find the minimum value of the difference on $1$.
Solution:
Minimum is at $x = 0$:
$f_2(0) = 0$, $g_2(0) = 1$
Difference: $1$
5. Error in Prediction (Area Between Curves)
Question:
Compute the error in prediction for company A (difference in area between $f_1(x) = x^3$ and $g_1(x) = \sqrt{x}$) and company B (difference in area between $f_2(x) = x$ and $g_2(x) = e^x$) over $1$.
