1. Consider the following.
f(x)=8x-10 g(x)=x^2-4x+10
(a) Find the points of intersection of the graphs.
ANSWER:(2,6) are the smaller values and (10,70) are the larder values
(b) Compute the area of the region below the graph of f and above the graph or g.
How do I answer part b?
You could draw a graph and count squares, or compute the integral of f(x) - g(x) between the intersection points
any points: I can use (2,6) or (10,70)?
To compute the area of the region below the graph of f(x) and above the graph of g(x), you will need to evaluate the definite integral of the difference between the two functions within a certain interval.
In this case, you can find the interval of intersection by setting f(x) equal to g(x) and solving for x.
1. Set f(x) = g(x):
8x - 10 = x^2 - 4x + 10
2. Rearrange the equation to get a quadratic equation set to zero:
x^2 - 12x + 30 = 0
3. Solve the quadratic equation either by factoring, completing the square, or using the quadratic formula. In this case, the equation does not factor nicely, so you can use the quadratic formula:
x = (-(-12) ± sqrt((-12)^2 - 4(1)(30))) / (2(1))
Simplifying gives two possible values for x:
x ≈ 1.601 and x ≈ 10.399
4. Calculate the definite integral of the difference between f(x) and g(x) within the interval of intersection:
∫[1.601, 10.399] (f(x) - g(x)) dx
You can evaluate this integral using standard integration techniques such as the Power Rule or the method of substitution.
Once you have the result, you will obtain the area of the region below the graph of f(x) and above the graph of g(x).