Re: differentaition word problem
Hello, Becky4paws!
From the way you went way off in your first step,
\displaystyle \;\; I assume you could use a walk-through . . .
Find all the points
( x , y ) \displaystyle (x,y) ( x , y ) on the graph of the function
y = 4 x 2 \displaystyle y\,=\,4x^2 y = 4 x 2
with the property that the tangent to the graph at
( x , y ) \displaystyle (x,y) ( x , y ) passes thru the point
( 2 , 0 ) . \displaystyle (2,0). ( 2 , 0 ) .
I don't suppose you made a sketch . . . [Naw, whatever for ??]
Code:
* | */
| /
| o
* | */
* | * /
-----------***----*------
| /(2,0)
| /
Here is one approach to this problem . . .
The line through
( 2 , 0 ) \displaystyle (2,0) ( 2 , 0 ) has slope
m . \displaystyle m. m .
Its equation is:
y − 0 = m ( x − 2 ) ⇒ y = m x − 2 m \displaystyle \,y\,-\,0\:=\:m(x\,-\,2)\;\;\Rightarrow\;\;y\:=\:mx\,-\,2m y − 0 = m ( x − 2 ) ⇒ y = m x − 2 m
Find the intersections of the parabola and this line:
\(\displaystyle \;\;4x^2\:=\:mx\,-\.2m\;\;\Rightarrow\;\;4x^2\,-\,mx\,+\,2m\:=\:0\)
Quadratic Formula:
x = − ( − m ) ± ( − m ) 2 − 4 ( 4 ) ( 2 m ) 2 ( 4 ) = m ± m 2 − 32 m 8 \displaystyle \,x\:=\:\frac{-(-m)\,\pm\,\sqrt{(-m)^2\,-\,4(4)(2m)}}{2(4)}\;= \;\frac{m\,\pm\,\sqrt{m^2\,-\,32m}}{8}\: x = 2 ( 4 ) − ( − m ) ± ( − m ) 2 − 4 ( 4 ) ( 2 m ) = 8 m ± m 2 − 3 2 m [1]
We want the line to be
tangent to the parabola,
\displaystyle \;\; so there will be exactly
one intersection.
This happens when the discriminant is zero:
m 2 − 32 m = 0 \displaystyle \,m^2\,-\,32m\:=\:0 m 2 − 3 2 m = 0
We have: \(\displaystyle \:m(m\,-\,32)\:=\0\;\;\Rightarrow\;\;m\:=\:0,\,32\)
Substitute
m = 0 \displaystyle m\,=\,0 m = 0 into
[1] :
x = 0 \displaystyle \,x\:=\:0 x = 0
\displaystyle \;\; Then:
y = 4 ⋅ 0 2 = 0 \displaystyle \,y\:=\:4\cdot0^2\:=\:0 y = 4 ⋅ 0 2 = 0
The line is tangent at
( 0 , 0 ) . \displaystyle (0,0). ( 0 , 0 ) .
Substitute
m = 32 \displaystyle m\,=\,32 m = 3 2 into
[1] :
x = 4 \displaystyle \,x\:=\:4 x = 4
\displaystyle \;\; Then:
y = 4 ⋅ 4 2 = 64. \displaystyle \,y\:=\:4\cdot4^2\:=\:64. y = 4 ⋅ 4 2 = 6 4 .
The line is tangent at
( 4 , 64 ) . \displaystyle (4,64). ( 4 , 6 4 ) .