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Let
f be the function given by
f(x)=4x^(2)-x^(3), and let
t be the line
y=18-3x, where
ℓ is tangent to the graph
prop+f. Let
R be the region bounded by the graph of
f and the
x-axis, and let
S be the region bounded by the graph of
f, the line
ℓ, and the
x-axis, as shown.
(a) Show that
ℓ is tangent to the graph of
y=f(x) at the point
x=3.
(b) Find the area of
S.
(c) Find the volume of the solid generated when
R is revolved about the
x-axis.

Let $f$ be the function given by $f(x)=4 x^{2}-x^{3}$ , and let $t$ be the line $y=18-3 x$ , where $\ell$ is tangent to the graph $\propto+f$ . Let $R$ be the region bounded by the graph of $f$ and the $x$ -axis, and let $S$ be the region bounded by the graph of $f$ , the line $\ell$ , and the $x$ -axis, as shown. $\newline$ (a) Show that $\ell$ is tangent to the graph of $f(x)=4 x^{2}-x^{3}$ $4$ at the point $f(x)=4 x^{2}-x^{3}$ $5$ . $\newline$ (b) Find the area of $S$ . $\newline$ (c) Find the volume of the solid generated when $R$ is revolved about the $x$ -axis.

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Find equations of tangent lines using limits

Full solution

Q. Let

f

be the function given by

f(x)=4 x^{2}-x^{3}

, and let

t

be the line

y=18-3 x

, where

\ell

is tangent to the graph

\propto+f

. Let

R

be the region bounded by the graph of

f

and the

x

-axis, and let

S

be the region bounded by the graph of

f

, the line

\ell

, and the

x

-axis, as shown.

\newline

(a) Show that

\ell

is tangent to the graph of

f(x)=4 x^{2}-x^{3}

4

at the point

f(x)=4 x^{2}-x^{3}

5

.

\newline

(b) Find the area of

S

.

\newline

(c) Find the volume of the solid generated when

R

is revolved about the

x

-axis.

Verify Tangent Line: Verify that the line $y = 18 - 3x$ is tangent to $f(x)$ at $x = 3$ . $\newline$ To show that $\ell$ is tangent to $f$ at $x = 3$ , we need to check two things: the value of $f(3)$ and $f'(3)$ (the derivative of $f$ at $x = 3$ ). $\newline$ Calculate $f(x)$ $0$ . $\newline$ Calculate the derivative $f(x)$ $1$ , then $f(x)$ $2$ . $\newline$ The equation of the tangent line at $x = 3$ using point-slope form is $f(x)$ $4$ , simplifying to $f(x)$ $5$ . $\newline$ Since $y = 18 - 3x$ matches this, $\ell$ is indeed tangent to $f$ at $x = 3$ .
Calculate Area of Region: Find the area of region $S$ . $\newline$ Region $S$ is bounded by $f(x)$ , $\ell$ , and the x-axis from $x = 0$ to $x = 3$ . $\newline$ Calculate the area under $f(x)$ from $x = 0$ to $x = 3$ using integration: $\int_0^3 (4x^2 - x^3) \, dx$ . $\newline$ This integral evaluates to $S$ $0$ . $\newline$ Calculate the area under $\ell$ from $x = 0$ to $x = 3$ : $S$ $4$ . $\newline$ This integral evaluates to $S$ $5$ . $\newline$ Area of $S$ is $S$ $7$ square units.
Find Volume of Solid: Find the volume of the solid generated when $R$ is revolved about the x-axis. $\newline$ Use the disk method to find the volume of the solid formed by revolving $R$ around the x-axis from $x = 0$ to $x = 3$ . $\newline$ Volume $V = \pi \int_0^3 (4x^2 - x^3)^2 \, dx$ . $\newline$ This integral is complex, but simplifies to $\pi \left[ \frac{4}{3}x^3 - \frac{1}{4}x^4 \right]_0^3$ squared. $\newline$ Evaluating this integral incorrectly leads to a wrong volume calculation.

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\newline

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\newline

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