Math

How do you determine the convergence or divergence of an infinite series using the RatioTo determine the convergence or divergence of an infinite series using the Ratio Test, compute the limit L = lim (n→∞) |a_(n+1) / a_n|. If L < 1, the series converges absolutely. If L > 1 or L is infinite, the series diverges. If L = 1, the test is inconclusive.

How do you prove that the angles opposite to each other in a cyclic quadrilateral are supplementary using the properties of a circle?In a cyclic quadrilateral, the sum of the opposite angles is supplementary because the measure of an angle subtended by an arc at the circumference is half the measure of the angle subtended by the same arc at the center. Therefore, the opposite angles sum to 180 degrees.

Can you explain how to simplify rational expressions involving polynomials?To simplify rational expressions involving polynomials, first factor both the numerator and the denominator completely. Next, identify and cancel any common factors between the numerator and the denominator. Finally, rewrite the expression with the remaining factors. It is crucial to check for any restrictions on the variable values that would make the original denominator zero.

How do you determine the limit of a function as it approaches a certain point using L’Hôpital’s Rule when direct substitution leads to an indeterminate form?To apply L’Hôpital’s Rule, first confirm the limit yields an indeterminate form like 0/0 or ∞/∞. Then, differentiate the numerator and denominator separately and re-evaluate the limit. Repeat if necessary until the limit can be determined or another method is required.

How do you calculate the standard deviation of a given data set?To calculate the standard deviation of a given data set, follow these steps: 1) Find the mean (average) of the data set. 2) Subtract the mean from each data point and square the result. 3) Find the average of these squared differences. 4) Take the square root of this average. This gives you the standard deviation, a measure of how spread out the numbers in your data set are.

How do you find the domain and range of a composite function?To find the domain of a composite function f(g(x)), determine the domain of g(x) and then find the values in this domain that make f(g(x)) valid. For the range, evaluate f(g(x)) using the domain of g(x) and find the resulting values.

What are the basic trigonometric functions and how do you use them to find the sides of a right triangle?The basic trigonometric functions are sine (sin), cosine (cos), and tangent (tan). They are defined as follows: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. To find the sides of a right triangle, you can rearrange these formulas to solve for the unknown side given one angle and one side.

How do you find the vertex of a quadratic function and determine if it represents a maximum or minimum point?To find the vertex of a quadratic function in the form y = ax^2 + bx + c, use the formula x = -b/(2a). Substitute this x-value back into the function to find the y-coordinate. If a > 0, the vertex is a minimum point; if a < 0, it is a maximum point.

How can you calculate the area of a triangle given the lengths of its three sides using Heron’s formula?To calculate the area of a triangle using Heron’s formula, first determine the semi-perimeter (s) by adding the lengths of the sides (a, b, c) and dividing by two: s = (a + b + c) / 2. Then, use the formula: Area = √[s(s – a)(s – b)(s – c)].