Math

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)].

How do you prove that the sum of the interior angles of a convex polygon with n sides is (n-2)*180 degrees using inductive reasoning?To prove this, use mathematical induction. Base case: For a triangle (n=3), the sum is 180 degrees. Inductive step: Assume true for n=k. For n=k+1, divide the polygon into a triangle and a k-sided polygon, proving the formula holds for n=k+1. Thus, by induction, the sum of interior angles of an n-sided convex polygon is (n-2)*180 degrees.

How do you solve trigonometric equations involving both sine and cosine within specific intervals and verify the solutions using unit circle principles?To solve trigonometric equations involving both sine and cosine within specific intervals, isolate one trigonometric function, use identities to simplify, and solve for the angle. Verify solutions by checking them on the unit circle, ensuring they lie within the given interval.

How do you solve the equation 3x – 4 = 11?To solve the equation 3x – 4 = 11, first add 4 to both sides to get 3x = 15. Then, divide both sides by 3 to find x = 5.

How do you compute the limit of a multivariable function using L’Hopital’s Rule when approaching the origin?To compute the limit of a multivariable function using L’Hopital’s Rule when approaching the origin, first confirm the limit is in an indeterminate form. Then, apply partial derivatives to each variable iteratively, simplifying the function. Repeat until the limit can be evaluated directly.

How do you solve systems of nonlinear equations using substitution and elimination methods?To solve systems of nonlinear equations using substitution, solve one equation for one variable and substitute into the other. For elimination, manipulate equations to cancel one variable. Both methods simplify the system to solve for all variables. Verify solutions by substituting back into original equations.

How do you prove that the diagonals of a parallelogram bisect each other using coordinate geometry?To prove that the diagonals of a parallelogram bisect each other using coordinate geometry, consider a parallelogram ABCD with vertices A(x1, y1), B(x2, y2), C(x3, y3), and D(x4, y4). The midpoint of diagonal AC is ((x1+x3)/2, (y1+y3)/2) and the midpoint of diagonal BD is ((x2+x4)/2, (y2+y4)/2). Since ABCD is a parallelogram, opposite sides are equal and parallel, leading to the conclusion that these midpoints are the same, thus proving that the diagonals bisect each other.

What is the power rule for differentiation, and can you provide a simple example of how it is used?The power rule for differentiation states that if you have a function f(x) = x^n, where n is a constant, then the derivative of the function is f'(x) = nx^(n-1). For example, if f(x) = x^3, then f'(x) = 3x^2.

How do I solve the inequality 3x – 2 ≤ 7?To solve the inequality 3x – 2 ≤ 7, first add 2 to both sides to get 3x ≤ 9. Then, divide both sides by 3 to isolate x, resulting in x ≤ 3. Therefore, the solution to the inequality is x ≤ 3.