Finding tangent lines, y-intercepts, x-intercepts, etc. with a given equation

[mathiscool]

Homework Statement

Let f(x)= a (7-x^2) for all a does not equal 0
a) Find, in terms of a, the equations of the lines tangent to these curves at x=-1
b) Find, in terms of a, the y-intercepts of the tangent lines at x=-1
c) find the x-intercepts of the tangent lines at x=-1
d)find, in terms of a, the area enclosed by the graph of f(x), the tangent line at x=-1, and the y-axis

Homework Equations

None? How to find a derivative?

The Attempt at a Solution

So I finished part a and i got y=2ax +8a
I got the derivative (-2ax) which was also the slope then i just plugged -1 into the original equation and I got 6a. Using the point slope equation, i got y = 2ax +8a, which i know is right. I don't understand b,c, or d so any help would be great! Thanks :)

 

[Mark44]

Homework Statement

Let f(x)= a (7-x^2) for all a does not equal 0
a) Find, in terms of a, the equations of the lines tangent to these curves at x=-1
b) Find, in terms of a, the y-intercepts of the tangent lines at x=-1
c) find the x-intercepts of the tangent lines at x=-1
d)find, in terms of a, the area enclosed by the graph of f(x), the tangent line at x=-1, and the y-axis

Homework Equations

None? How to find a derivative?

The Attempt at a Solution

So I finished part a and i got y=2ax +8a
I got the derivative (-2ax) which was also the slope then i just plugged -1 into the original equation and I got 6a. Using the point slope equation, i got y = 2ax +8a, which i know is right. I don't understand b,c, or d so any help would be great! Thanks :)

I hope that you are drawing a sketch of your function. Actually, two sketches would be better - one that assumes a positive value for a, and another that assumes a negative value for a. Both sketches should show a parabola, with one opening up and the other opening down.

On each sketch, draw a tangent line at the point (-1, y). You have the derivative function, so you should be able to get the slope of the tangent line(s). You also have the y value on each graph, so you should be able to get the equation of the tangent line(s). Once you have the tangent line equation(s), it should be a simple matter to find the y-intercept(s) of the tangent line(s). That's part b.

For part c, use the tangent line equation(s) to determine where they cross the x-axis.

Once you get parts b and c, we can talk about part d.

 

[mathiscool]

Thanks! I got x=-4 for part c and now I'm down to part D. I think I'm supposed to set the original equal to the equation and solve in terms of a, but from that I'm getting a value for a. Do i plug that value, along with x=-1 into the original to find the area?

 

[Mark44]

I haven't worked this problem, but I don't believe you should get a numeric value for part c. The expression you end up with should involve a, I believe.

I think I'm supposed to set the original equal to the equation and solve in terms of a, but from that I'm getting a value for a. Do i plug that value, along with x=-1 into the original to find the area?

I have no idea what you're saying here. I'm looking for a single word that would indicate you're at least thinking in the right direction, and I don't see it.

How do you normally find the area of some region?