Area between a function and its tangent

In summary, to find the area of the region bounded by the graph of f(x) = 4x^2, the tangent line at P(2, f(2)), and the x-axis, one can find the equation for the tangent line by taking the derivative and substituting the x-value of 2. However, it may be easier to simply integrate under the parabola and subtract the area of the triangle formed by the tangent line.
  • #1
Momentum09
71
0

Homework Statement


Find the area of the region bounded by the graph of f(x) = 4x^2, the tangent line to this graph at P(2, f(2)), and the x-axis


Homework Equations



Integral of [f(x)-g(x) dx]


The Attempt at a Solution


I first tried to find the equation for the tangent line
The derivative of 4x^2 = 8x, subsituting x=2 into the equation I got 16 [slope]
the equation then turned out to be y-2 = 16 (x-2) --> y = 16x-30
I then subtracted this tangent equation from 4x^2 then integrate, but I wasn't able to get the right answer.
Please show me how...thank you!
 
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  • #2
Why did you have y - 2 = 16(x - 2)? If I plug in x = 2 I think it is not a tangent line...
 
  • #3
Hi Momentum09! :smile:

I'll just add to what CompuChip :smile: says …

why not just integrate under the parabola in the usual way, and then subtract the area of the triangle using half-base-times-height? :wink:
 

1. What is the definition of the area between a function and its tangent?

The area between a function and its tangent is the region bounded by the curve of the function and its tangent line at a specific point of intersection.

2. How is the area between a function and its tangent calculated?

The area between a function and its tangent can be calculated using calculus, specifically by finding the definite integral of the function from the point of intersection to another point on the curve.

3. What is the significance of the area between a function and its tangent?

The area between a function and its tangent is related to the concept of instantaneous rate of change. It represents the amount of change in the function at a specific point, and can be used to analyze the behavior of the function.

4. Can the area between a function and its tangent be negative?

Yes, the area between a function and its tangent can be negative. This can occur when the function is decreasing at the point of intersection, resulting in a negative area under the curve.

5. How does the value of the area between a function and its tangent change as the point of intersection moves?

The value of the area between a function and its tangent can change depending on the location of the point of intersection. As the point of intersection moves, the area can increase, decrease, or remain constant, depending on the behavior of the function at that point.

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