What is the equation of the line that

Raven Gardner

Raven Gardner

Answered question

2022-04-09

What is the equation of the line that is normal to f(x)=x35x+2 at x=3?

Answer & Explanation

Egerlandsq0z

Egerlandsq0z

Beginner2022-04-10Added 14 answers

Explanation:
Take the derivative of f(x). The derivative of f(x) is
f(x)=3(x2)5
Plug 3 into the derivative to get
f(3)=3(32)5=22
That is the slope tangent to the curve but to find the slope of the tangent line, you need to take the reciprocal of that number and then switch signs, so the slope of the normal line is 122.
Now plug 3 into the initial equation.
f(3)=335(3)+2=14
Now that we have the point (3, 14) and the slope of the normal line, we can plug into the equation
(yy1)=dydx(xx1)
Plug in 14 for y1,122 for dydx, and 3 for x1. Solve for y and you should get
y=(x22)+(4522)
Jax Burns

Jax Burns

Beginner2022-04-11Added 13 answers

We are given that f(x)=x35x+2 and x0=3.
Find the value of the function at the given point: y0=f(3)=14.
The slope of the normal line at x=x0 is the negative reciprocal of the derivative of the function, evaluated at x=x0:M(x0)=1f(x0)
Find the derivative: f(x)=(x35x+2)=3x25
Hence, M(x0)=1f(x0)=13x025
Next, find the slope at the given point.
m=M(3)=122
Finally, the equation of the normal line is yy0=m(xx0)
Plugging the found values, we get that y14=x322
Or, more simply: y=31122x22.

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