The line has equation y=2x+c and a curve has equation y=8-2x-x^2.
1) for the case where the line is a tangent to the curve, find the value of the constant c.
2) For the case where c = 11, find the x-coordinates of the points of intersection of the line and the curve. Find also, by integration, the area of region between the line and the curve.
to solve:
2x+c = 8-2x-x^2
x^2 + 4x + c-8 = 0
to have y = 2x+c to be a tangent, there can be only one real root of the quadratic
so b^2 - 4ac = 0
16 - 4(1)(c-8) = 0
16 - 4c + 32 = 0
4c = 48
c = 48/4 = 12
check with Wolfram:
http://www.wolframalpha.com/input/?i=y%3D2x%2B12,+y%3D8-2x-x%5E2
for your second part
y = 2x+11, y = 8-2x - x^2
I will let you solve it to show:
http://www.wolframalpha.com/input/?i=y%3D2x%2B11,+y%3D8-2x-x%5E2
so for the area:
A = ∫ (8-2x-x^2 - 2x - 11) dx from -3 to -1
simplify and finish it
this looks very straightforward, I trust you can handle it
No it isnt that straightforward u must take into account the area of the trapezium as well so the best thing to do is to draw out the diagram
1) Well, well, we have a tangential situation here! To find the value of the constant c when the line is a tangent to the curve, we need to set the equations equal to each other. So, let's equate them:
2x + c = 8 - 2x - x^2
Now, we need to solve this equation for x. Are you ready? Here's the fun part:
x^2 - 4x + (c - 8) = 0
Now, for the line to be a tangent to the curve, this equation should have only one solution, right? In other words, the discriminant (b^2 - 4ac) should be equal to zero. So, let's calculate that:
(4^2) - 4(1)(c - 8) = 0
16 - 4c + 32 = 0
-4c + 48 = 0
-4c = -48
c = 12
Voilà! The value of the constant c is 12.
2) Ah, now it's getting interesting! When c = 11, we need to find the x-coordinates of the points of intersection between the line and the curve. To do that, we'll set the equations equal to each other again:
2x + 11 = 8 - 2x - x^2
Now, let's solve this equation for x. Brace yourself!
x^2 - 4x + 3 = 0
Haha, it turns out to be a convenient quadratic equation. We can easily factorize it:
(x - 3)(x - 1) = 0
So, either x - 3 = 0 or x - 1 = 0. That means x can be either 3 or 1. Bingo!
Now, for the area between the line and the curve, we need to find the definite integral. Let's integrate the curve equation from x = 1 to x = 3:
∫(8 - 2x - x^2)dx
I'll spare you the math, but after solving the integral, we get the area between the line and the curve as a result. So, I guess it's time for a drumroll!
*Drumroll*
The area between the line and the curve, when c = 11, is... uh... some number!
To find the value of the constant c when the line is a tangent to the curve, we need to equate their slopes because the slope of the line is equal to the derivative of the curve at the point of tangency.
1) To find the derivative of the curve, we differentiate the equation y = 8 - 2x - x^2 with respect to x:
dy/dx = -2 - 2x
The slope of the line is 2, which means:
2 = -2 - 2x
Rearranging the equation, we get:
2x = -2 - 2
2x = -4
x = -2
Substituting the value of x into either the line's equation or the curve's, we can find the corresponding y-value:
y = 2x + c
y = 2(-2) + c
y = -4 + c
Now, we equate the y-values:
8 - 2x - x^2 = -4 + c
Rearranging the equation, we obtain a quadratic equation:
x^2 + 2x + (c - 12) = 0
Since the line is tangent to the curve, this quadratic equation will have only one solution. For a quadratic equation to have only one solution, the discriminant must be zero:
b^2 - 4ac = 0
In our case, a = 1, b = 2, and c = (c - 12). Substituting these values into the discriminant formula:
(2)^2 - 4(1)(c - 12) = 0
4 - 4c + 48 = 0
-4c + 52 = 0
-4c = -52
c = 13
Therefore, the value of the constant c when the line is a tangent to the curve is 13.
2) For the case where c = 11, we can find the x-coordinates of the points of intersection by equating the line and the curve:
2x + 11 = 8 - 2x - x^2
Rearranging the equation, we have:
x^2 + 4x - 3 = 0
Using the quadratic formula, we find:
x = (-4 ± √(4^2 - 4(1)(-3))) / (2(1))
x = (-4 ± √(16 + 12)) / 2
x = (-4 ± √28) / 2
x = (-4 ± 2√7) / 2
Simplifying further, we get:
x = -2 ± √7
Therefore, the x-coordinates of the points of intersection are -2 + √7 and -2 - √7.
To find the area of the region between the line and the curve, we need to integrate the difference between the line and the curve across the interval of x-values where they intersect.
The area A can be calculated using the definite integral:
A = ∫[a,b] (f(x) - g(x)) dx
In this case, f(x) represents the equation of the line (2x + 11) and g(x) represents the equation of the curve (8 - 2x - x^2), with limits of integration from -2 - √7 to -2 + √7.
A = ∫[-2 - √7, -2 + √7] ((2x + 11) - (8 - 2x - x^2)) dx
Integrating and evaluating, we find:
A = [x^2 + 11x + 8x - (8x - x^2 - (1/3)x^3)]|[-2 - √7, -2 + √7]
Simplifying further, we get:
A = [(1/3)x^3 + 19x - (8/3)x^3 + 11x + 8x]|[-2 - √7, -2 + √7]
Evaluating the definite integral, we find the area of the region between the line and the curve when c = 11.