Solved on Oct 23, 2023

Use a linear approximation to estimate $(3.01)^{3}$ .

STEP 1

Assumptions1. We are asked to use a linear approximation to estimate the quantity $(3.01)^3$ . . We know that a linear approximation of a function $f(x)$ at a point $a$ is given by the tangent line to the function at that point, which can be represented as $f(a) + f'(a)(x-a)$ .
3. We can choose a function $f(x) = x^3$ and a point $a =3$ for our linear approximation, because $3^3$ is easy to calculate and close to our desired value $(3.01)^3$ .

STEP 2

First, we need to find the derivative of the function f(x) = x^. This can be done using the power rule, which states that the derivative of $x^n$ is $nx^{n-1}$ .
$f'(x) =x^2$

STEP 3

Now, plug in the value $a =3$ into the derivative to find the slope of the tangent line at the point $(3, f(3))$ .
$f'(3) =3(3)^2$

STEP 4

Calculate the value of the derivative at $a =3$ .
$f'(3) =3(3)^2 =27$

STEP 5

Now, we can write the equation for the linear approximation of the function $f(x)$ at the point $a =3$ .
$(x) = f(3) + f'(3)(x -3)$

STEP 6

Plug in the values for $f(3)$ and $f'(3)$ into the equation for the linear approximation.
$(x) =3^3 +27(x -3)$

STEP 7

implify the equation for the linear approximation.
$(x) =27 +27(x -3)$

STEP 8

Now, we can use the linear approximation to estimate the value of $(3.01)^3$ . Plug in $x =3.01$ into the equation for the linear approximation.
$(3.01) =27 +27(3.01 -3)$

STEP 9

implify the equation to find the estimate.
$(3.01) =27 +27(.01) =27 +.27 =27.27$ So, using a linear approximation, we estimate that $(3.01)^3 \approx27.27$ .

Was this helpful?

Use a linear approximation to estimate $(3.01)^{3}$ .

STEP 1

STEP 2

First, we need to find the derivative of the function f(x) = x^. This can be done using the power rule, which states that the derivative of $x^n$ is $nx^{n-1}$ .
$f'(x) =x^2$

STEP 3

Now, plug in the value $a =3$ into the derivative to find the slope of the tangent line at the point $(3, f(3))$ .
$f'(3) =3(3)^2$

STEP 4

Calculate the value of the derivative at $a =3$ .
$f'(3) =3(3)^2 =27$

STEP 5

Now, we can write the equation for the linear approximation of the function $f(x)$ at the point $a =3$ .
$(x) = f(3) + f'(3)(x -3)$

STEP 6

Plug in the values for $f(3)$ and $f'(3)$ into the equation for the linear approximation.
$(x) =3^3 +27(x -3)$

STEP 7

implify the equation for the linear approximation.
$(x) =27 +27(x -3)$

STEP 8

Now, we can use the linear approximation to estimate the value of $(3.01)^3$ . Plug in $x =3.01$ into the equation for the linear approximation.
$(3.01) =27 +27(3.01 -3)$

STEP 9

implify the equation to find the estimate.
$(3.01) =27 +27(.01) =27 +.27 =27.27$ So, using a linear approximation, we estimate that $(3.01)^3 \approx27.27$ .

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Use a linear approximation to estimate (3.01)3(3.01)^{3}(3.01)3.

STEP 1

STEP 2

First, we need to find the derivative of the function f(x) = x^. This can be done using the power rule, which states that the derivative of xnx^nxn is nxn−1nx^{n-1}nxn−1.f′(x)=x2f'(x) =x^2f′(x)=x2

STEP 3

Now, plug in the value a=3a =3a=3 into the derivative to find the slope of the tangent line at the point (3,f(3))(3, f(3))(3,f(3)).f′(3)=3(3)2f'(3) =3(3)^2f′(3)=3(3)2

STEP 4

Calculate the value of the derivative at a=3a =3a=3.f′(3)=3(3)2=27f'(3) =3(3)^2 =27f′(3)=3(3)2=27

STEP 5

Now, we can write the equation for the linear approximation of the function f(x)f(x)f(x) at the point a=3a =3a=3.(x)=f(3)+f′(3)(x−3)(x) = f(3) + f'(3)(x -3)(x)=f(3)+f′(3)(x−3)

STEP 6

Plug in the values for f(3)f(3)f(3) and f′(3)f'(3)f′(3) into the equation for the linear approximation.(x)=33+27(x−3)(x) =3^3 +27(x -3)(x)=33+27(x−3)

STEP 7

implify the equation for the linear approximation.(x)=27+27(x−3)(x) =27 +27(x -3)(x)=27+27(x−3)

STEP 8

Now, we can use the linear approximation to estimate the value of (3.01)3(3.01)^3(3.01)3. Plug in x=3.01x =3.01x=3.01 into the equation for the linear approximation.(3.01)=27+27(3.01−3)(3.01) =27 +27(3.01 -3)(3.01)=27+27(3.01−3)

STEP 9

implify the equation to find the estimate.(3.01)=27+27(.01)=27+.27=27.27(3.01) =27 +27(.01) =27 +.27 =27.27(3.01)=27+27(.01)=27+.27=27.27So, using a linear approximation, we estimate that (3.01)3≈27.27(3.01)^3 \approx27.27(3.01)3≈27.27.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Use a linear approximation to estimate $(3.01)^{3}$ .

First, we need to find the derivative of the function f(x) = x^. This can be done using the power rule, which states that the derivative of $x^n$ is $nx^{n-1}$ .
$f'(x) =x^2$

Now, plug in the value $a =3$ into the derivative to find the slope of the tangent line at the point $(3, f(3))$ .
$f'(3) =3(3)^2$

Calculate the value of the derivative at $a =3$ .
$f'(3) =3(3)^2 =27$

Now, we can write the equation for the linear approximation of the function $f(x)$ at the point $a =3$ .
$(x) = f(3) + f'(3)(x -3)$

Plug in the values for $f(3)$ and $f'(3)$ into the equation for the linear approximation.
$(x) =3^3 +27(x -3)$

implify the equation for the linear approximation.
$(x) =27 +27(x -3)$

Now, we can use the linear approximation to estimate the value of $(3.01)^3$ . Plug in $x =3.01$ into the equation for the linear approximation.
$(3.01) =27 +27(3.01 -3)$

implify the equation to find the estimate.
$(3.01) =27 +27(.01) =27 +.27 =27.27$ So, using a linear approximation, we estimate that $(3.01)^3 \approx27.27$ .