Determine Equation of the the lines tan to...

[krete77]

The problem is:
Tangent Lines: Determine equations of the lines tangent to the graph of Y = x?(5-x²) at the points (1,2) and (-2,-2). Graph the function and the tangent lines.

I have no IDEA where to go with this. I am taking calculus over the summer and we are in week 2 and I'm struggling..if anyone could do a step by step process here and explain I would be so grateful. Thanks in advance.

 

[galactus]

I step through one of them and you try the other..Okey-doke?.

Find tangent line to \(\displaystyle y=x\sqrt{5-x^{2}}\) at (1,2):

We can use y=mx+b

We are given x and y, so we need to find m and b.

m is the slope of the line at said point. We find this through the derivative.

Product rule and chain rule:

\(\displaystyle m=y'(x)=x\left(\frac{1}{2}(5-x^{2})^{\frac{-1}{2}}\right)(-2x)+\sqrt{5-x^{2}}(1)\)

\(\displaystyle =\sqrt{5-x^{2}}-\frac{x^{2}}{\sqrt{5-x^{2}}}=\frac{5-2x^{2}}{\sqrt{5-x^{2}}}\)

So, sub x=1 into the derivative and we get \(\displaystyle m=\frac{3}{2}\)

Plug it all into y=mx+b and solve for b, x=1, y=2, m=3/2

\(\displaystyle 2=\frac{3}{2}(1)+b\Rightarrow b=1/2\)

The line equation is \(\displaystyle y=\frac{3}{2}x+\frac{1}{2}\)

The worst part for you may be finding the derivative and the algebra involved. The product rule and chain rule are needed for this one. The graphing can be done easily with a calculator or other graphing utility. If you need a nice FREE graphing utility, go here:

http://www.padowan.dk