1
$\begingroup$

For example, find a curve that is tangent to the line $y=2x-3$ at $x = 5$. Is the solution (curve) unique?

  • 0
    I'm not sure what you mean exactly. If the line is the tangent line at that point, the curve would be found by integrating the line, wouldn't be? – 2017-02-19
  • 0
    @Masacroso Oh! That makes much more sense... – 2017-02-19

1 Answers 1

3

The derivative at $x=5$ is $2$. Then the only restriction that our curve must have is that at the point $(5,7)$ it must have instantaneous slope $2$. So the only restriction is that,

$$f(5)=7$$

$$f'(5)=2$$

The solution is not unique.

For example take $f(x)=ax^2+bx+c$ with $f(5)=25a+5b+c=7$ and $f'(5)=2a(5)+b=2$. Two linear equations, 3 unknowns. Thus an infinite amount of solutions just for quadratics, and we can likely find others.

If you want only one solution, looking at what we have above, maybe let $a=1$ so that $b=-8$ and $c=22$.

So one solution is,

$$f(x)=x^2-8x+22$$

  • 0
    Got it. Thank you for your guidance. – 2017-02-19
  • 0
    You're welcome! – 2017-02-19