Question

CALCULUS : NORMAL CURVE

Suppose \(f(x)= -\frac{ 1 }{ 4 }x^2+3x+4\).
a. i. Write down \(f'(x)\) in simplest form.
ii. Find the equation of the normal to \(f\) at \((2,9)\).
iii. If this normal cuts \(f\) at another point A, find the coordinates of A.

Answer 1

Nice that you've used Equation Editor. Neat, accurate results!

What's the derivative of 4?

What's the derivative of 3x?

Answer 2

Hold on; my question is not finished. OS is so slow. -_-

Answer 3

a. i. \(f'(x)=-\frac{ 1 }{ 2 }x+3\)

Answer 4

Correct?

Answer 5

@mathmale

Answer 6

Yeah, power rule　

Answer 7

All right. So for ii., how should I start?

Answer 8

ii)

step 1) plug in x = 2 into the f ' (x) equation. Evaluate to find the value of the slope of the tangent line. Call it `m`

step 2) Compute `-1/m` to find the perpendicular slope

Step 3) use either point slope or slope intercept form to find the equation of the normal line

Answer 9

\(f'(2)=2 \) therefore m= -1/2

Answer 10

y-9=-1/2(x-2)

or

y=-1/2+10

Answer 11

right?

Answer 12

the second equation I think you meant to say `y = (-1/2)x+10`

Answer 13

if so, then they both look good. I would go with \(\Large y = -\frac{1}{2}x+10\)

Answer 14

yes, i forgot the x :)

Answer 15

then for iii., do i just equate the f(x) and f'(x)?

Answer 16

if so, i got (12, 4) as coordinates

Answer 17

you have the system of equations with f and the normal line

so these two equations

y = -1/4x^2+3x+4
y = -1/2x + 10

solve this system. One of the points will be (2,9). They want you to find the other point

Answer 18

oh i meant the normal curve and the function not the derivative

Answer 19

i'll show my work

Answer 20

(12,4) is correct. That is the other point

Answer 21

okay thanks! :)

Answer 22

no problem