There are no numbers, but don't panic. This will work the same way as all the other problems. We need a point and a slope. Since we don't have a formula, we can't figure out what *f*(*a*) is, but we know the point (*a*, *f*(*a*)) is the one we want. Since we don't have a formula, we can't figure out what
*f ' *(*a*) is, but we know *f ' *(*a*) is the slope we want. We want a line of the form *y* = *mx* + *b*
whose slope is *f ' *(*a*), so* m* = *f *' (*a*): *y* = *f* ' (*a*)*x* + *b*.
We also want (*a*, *f*(*a*)) to be on the line, so *f*(*a*) = *f *' (*a*)(*a*) + *b*.
Solving, we find *b* = *f*(*a*) – *f *' (*a*)*a*. This means the equation for the tangent line to *f* at *a* is
*y* = *f *' (*a*)*x* + (*f*(*a*) – *f *' (*a*)*a*). This is a little messy. We can make it prettier by rearranging. First, remove the unnecessary parentheses: *y* = *f* ' (*a*)*x* + *f*(*a*) – *f *' (*a*)*a*
Then swap the second and third terms: *y* = *f *' (*a*)*x* – *f *' (*a*)*a* + *f*(*a*)
Pull out *f ' *(*a*) from the first two terms: And there we are. The magic formula says the equation for the tangent line is *y* = *f* ' (*a*)(*x* – *a*) + *f*(*a*).
We might also see it written *y* = *f*(*a*) + *f *' (*a*)(*x* – *a*),
which is the same thing with the terms switched. Since we've given in and explained the magic formula, we should probably show how to use it, too. Here, we aren't as nice. We'll need to find the derivatives from scratch, using the sneaky methods. |