Calculus Math Question!

Question

Question: The line through A(2,4) and B(0,8) is a tangent to y= a/(x+2)². Find a.

Answer: a=64

Answer 1

There are three parts to solving this problem.

Part 1

Whenever you're stuck, it's always nice to start with the easiest thing possible :). So let's first figure out the equation of the line that they gave us.

It passes through the points A(2,4) and B(0,8), so the slope of this tangent line is: m = (8 - 4)/(0 - 2) = 4/-2 = -2

Very nicely, the point B is the y-intercept of the line, so now we know that the equation of the line is: y = -2x + 8

Remember, if this line is the tangent line to the given function at some point (x, y) on the graph, that means that the derivative of the function at some point (m, n) equals -2 (the slope of the tangent line.

In other words: y'(m) = -2

So let's take the derivative of the function y= a/(x+2)²

It may help if you rewrite the equation as: y= a(x+2)^-2

Using the power rule (and technically also the chain rule, though it's unnecessary), the derivative is: y' = -2a(x+2)^-3 = (-2a)/(x+2)³

Now, we're saying that the slope of the tangent line at some point (m, n) equals -2, so:

(-2a)/(m+2)³ = -2

a/(m+2)³ = 1

a = (m + 2)³

Now, it looks like we're stuck again... so let's see if there's anything else we know.

Part 2:

If this line is the tangent line to the given function at some point (m, n) on the graph, that means that the point (m, n) is on the tangent line AND the original function. Since y = a/(x+2)², we can rewrite that as saying (m, a/(m+2)²) should also be a point on this tangent line.

So let's substitute this point in our tangent line equation:

a/(m+2)² = -2m + 8

(m + 2)³/(m + 2)² = -2m + 8

(m + 2) = -2m + 8

3m - 2 = 8

3m = 6

m = 2

And then, using our tangent line equation, n = -2(2) + 8 = 4

So that means that our tangent line goes through (2, 4). It also means that the point (2, 4) is on our function!

Part 3:

We can now figure out the value for "a"! Let's substitute (2, 4) into our equation: y= a/(x+2)²

4 = a/(2 + 2)²

4 = a/16

a = 64

And that's how we get a = 64.

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