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3 years ago

14

Please help. this is a final exam question

1 answer:

Varvara68 [4.7K]3 years ago

8 0

Step-by-step explanation:

Tanx =opp/adj

=50/42 =55. 5°

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Find the equation of the tangent line to the curve y equals left parenthesis x minus 3 right parenthesis superscript 7 baseline

Rashid [163]

We have been given the expression to be $y=(x-3)^7(x+1)^2$

Since we need to find the tangent at a point, we will have to find the derivative of $y$ as the slope of the tangent at a given point on the curve is always equal to value of the derivative at that point.

Thus, we have to find $\frac{dy}{dx}=\frac{d}{dx} (x-3)^7(x+1)^2$

We will use the product rule of derivatives to find $\frac{dy}{dx}$

Thus, $y'=7(x-3)^6(x+1)^2+(x-3)^7\times2(x+1)$ (using the product rule which states that $(fg)'=f'g+fg'$ )

Taking the common factors out we get:

$y'=(x-3)^6(x+1)(7(x+1)+2(x-3))$

$y'=(x-3)^6(x+1)(7x+7+2x-6)=(x-3)^6(x+1)(9x+1)$

Thus, $y'$ at $x=4$ is given by:

$y'_{x=4}$ =Slope of the tangent of y at x=4= $m$

Thus, $m=(4-3)^6(4+1)(9\times4+1)=185$

Now, the equation of the tangent line which passes through $(x_{1}, y_{1})$ and has slope m is given by:

$y-y_{1}=m(x-x_{1})$

Thus, the equation of the tangent line which passes through $(4,25)$ and has the slope 185 is $y-25=185(x-4)$

Which can be simplified to $y=185x-740+25=185x-715$

Thus, $y=185x-715$

This is the required equation of the tangent.

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3 years ago