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Let
f be the function given by
f(x)=4x^(2)-x^(3), and let
ℓ be the line
y=18-3x, where
ℓ is tangent to the graph of
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 $\ell$ be the line $y=18-3 x$ , where $\ell$ is tangent to the graph of $f$ . 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

\ell

be the line

y=18-3 x

, where

\ell

is tangent to the graph of

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.

Calculate $f'(x)$ : To verify if $\ell$ is tangent to $f$ at $x=3$ , calculate $f'(x)$ and evaluate it at $x=3$ .
Find intersection points: Find the points of intersection between $f(x)$ and $\ell$ to determine the bounds for the area of $S$ .
Calculate area of S: Calculate the area of S by integrating the difference between $f(x)$ and $\ell$ from $x = -2$ to $x = 3$ .

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