1. function g(x)=-x^2-3x+7 . Find the area under the curve for the domain –4£ x £ 1.
a 71.6 square units
b. 46 square units
c.35.83 square units
d.17.916square units
2. y=2sin(circle with a line through it)
a. How many cycles occur in the graph?
b. Find the period of the graph.
c. Find the amplitude of the graph.
did you ever find out the answer to 1??
Nope
1. To find the area under the curve of a function, we need to integrate the function over the given domain. In this case, the function is g(x) = -x^2 - 3x + 7 and the domain is -4 ≤ x ≤ 1.
a. Start by finding the antiderivative of g(x). The integral of -x^2 is -1/3 * x^3, the integral of -3x is -3/2 * x^2, and the integral of 7 is 7x.
So, the antiderivative of g(x) is G(x) = -1/3 * x^3 - 3/2 * x^2 + 7x.
b. To find the area under the curve for the given domain, we need to evaluate G(x) at the upper and lower limits of the domain and then find the difference:
Area = G(1) - G(-4)
= [-1/3 * (1)^3 - 3/2 * (1)^2 + 7(1)] - [-1/3 * (-4)^3 - 3/2 * (-4)^2 + 7(-4)]
Calculating this further, we get:
Area = [(-1/3) - (3/2) + 7] - [-(-64/3) - 3 * 16 + (-28)]
Simplifying this further, we get:
Area = [29/3 + 2 + 7] - [64/3 - 48 - 28]
Area = [54/3] - [64/3 - 76/3]
Area = 54/3 - (-12/3)
Area = 54/3 + 12/3
Area = 66/3 = 22
Therefore, the area under the curve for the given domain is 22 square units.
2. The given function is y = 2sin(θ), where θ represents the angle.
a. To find the number of cycles that occur in the graph, we need to determine the range of values of θ for which the function completes one cycle. In this case, the function is sin(θ), which completes one cycle from 0 to 2π (or 0 to 360 degrees).
Since the coefficient of sine is 2, the graph will complete 2 cycles within the interval 0 to 2π. Therefore, the answer is 2 cycles.
b. The period of a sine function is the length of one complete cycle. In this case, the period is 2π (360 degrees) divided by the coefficient of sine, which is 2.
Period = 2π / 2 = π
Therefore, the period of the graph is π.
c. The amplitude of a sine function is the distance from the midline to the maximum (or minimum) value of the function. In this case, the coefficient of sine is 2, which indicates that the amplitude is 2.
Therefore, the amplitude of the graph is 2.