What is the equation of the tangent line of #f(x)=6x-x^2 # at #x=-1#?
3 Answers
See below:
Explanation:
First step is finding the first derivative of
Hence:
The value of 8's significance is that this is the gradient of
So our line function is currently
However, we must also find the y-intercept, but to do this, we also need the y coordinate of the point where
Plug
So a point on the tangent line is
Now, using the gradient formula, we can find the equation of the line:
gradient
Hence:
Explanation:
We are given
To find the equation of the tangent line, we need to: find the slope of the tangent line, obtain a point on the line, and write the tangent line equation.
To find the slope of the tangent line, we take the derivative of our function.
Substituting our point
Now that we have our slope, we need to find a point on the line. We have an
So the point on the line is
With a slope and a point on the line, we can solve for the equation of the line.
Hence, the tangent line equation is:
Explanation:
#"we require the slope m and a point "(x,y)" on the line"#
#•color(white)(x)m_(color(red)"tangent")=f'(-1)#
#rArrf'(x)=6-2x#
#rArrf'(-1)=6+2=8#
#"and "f(-1)=-6-1=-7rArr(-1,-7)#
#rArry+7=8(x+1)#
#rArry=8x+1larrcolor(red)"equation of tangent"#