Math

How do you find the area between two curves using integration?To find the area between two curves, you first identify the points of intersection. Then, integrate the difference between the top function and the bottom function over the interval defined by these points. Mathematically, this is expressed as ∫[a,b] (f(x) – g(x)) dx, where f(x) is the upper curve and g(x) is the lower curve.

How do I solve the equation 3(x + 2) equals 18 and what properties should I use to simplify the expression?To solve the equation 3(x + 2) = 18, first use the Distributive Property to expand it to 3x + 6 = 18. Then, apply the Subtraction Property of Equality to isolate the variable: 3x = 12. Finally, use the Division Property of Equality to solve for x: x = 4.

How do you find the general solution for the trigonometric equation sin(x) = 1/2 in terms of degrees and radians?To find the general solution for sin(x) = 1/2, we identify the specific angles where this is true. In degrees, x = 30° + 360°n or x = 150° + 360°n, where n is any integer. In radians, x = π/6 + 2πn or x = 5π/6 + 2πn, where n is any integer.

How can I use the angle sum and difference identities to simplify the expression sin(75°)cos(15°) + cos(75°)sin(15°)?You can use the angle sum identity for sine, which states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Here, A = 75° and B = 15°, so the expression sin(75°)cos(15°) + cos(75°)sin(15°) simplifies to sin(90°), which equals 1.

How do you conduct and interpret hypothesis tests for two population means to determine a significant difference in statistics?To conduct hypothesis tests for two population means, first state the null hypothesis (H0) that the means are equal and the alternative hypothesis (H1) that they are not. Choose a significance level (α), collect sample data, and calculate the test statistic (e.g., t-test or z-test). Compare the test statistic to critical values or use the p-value approach. If the test statistic exceeds the critical value or the p-value is less than α, reject H0. Interpret results in context, considering effect size and practical significance.

How do you find the slope of a line using the coordinates of two points?To find the slope of a line using the coordinates of two points, use the formula: slope (m) = (y2 – y1) / (x2 – x1), where (x1, y1) and (x2, y2) are the coordinates of the two points. This formula calculates the change in the y-coordinates divided by the change in the x-coordinates.

What is the derivative of the function f(x) = 3x^2 + 2x + 1 with respect to x?The derivative of the function f(x) = 3x^2 + 2x + 1 with respect to x is found by applying the power rule. The derivative f'(x) = d/dx (3x^2) + d/dx (2x) + d/dx (1) results in f'(x) = 6x + 2.

What is the value of x if 3(x + 2) = 18?To find the value of x, first divide both sides by 3 to get x + 2 = 6. Then, subtract 2 from both sides to obtain x = 4.

How do you solve a system of linear equations using the matrix method (Gauss-Jordan elimination)?To solve a system of linear equations using the Gauss-Jordan elimination method, first express the system in matrix form as an augmented matrix. Then, perform row operations to transform the matrix into reduced row echelon form (RREF). Once in RREF, the solutions to the system can be read directly from the matrix.

How do you calculate the greatest common factor (GCF) of two numbers?To calculate the greatest common factor (GCF) of two numbers, list their factors, find the common factors, and select the largest one. Alternatively, use the Euclidean algorithm: divide the larger number by the smaller one, then replace the larger number with the remainder. Repeat until the remainder is zero; the last non-zero remainder is the GCF.