Math

What is the derivative of the function f(x) = x^2 and how do you find it?The derivative of the function f(x) = x^2 is found using basic differentiation rules. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). Applying this, the derivative of x^2 is 2x.

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 do you use the Pythagorean theorem to find the length of a side in a right triangle?To find a side length using the Pythagorean theorem, identify the lengths of two sides. For legs ‘a’ and ‘b’, and hypotenuse ‘c’, the formula is a² + b² = c². Rearrange to solve for the unknown side: a² = c² – b² or b² = c² – a², then take the square root.

How do you determine the area under the curve of the function f(x) using integration?To determine the area under the curve of the function f(x) using integration, you calculate the definite integral of f(x) over a given interval [a, b]. This is represented as ∫[a, b] f(x) dx. The result gives the total area between the curve and the x-axis within the specified interval.

How do you find the greatest common divisor (GCD) of two numbers?To find the greatest common divisor (GCD) of two numbers, use the Euclidean algorithm: divide the larger number by the smaller one, take the remainder, and repeat with the smaller number and the remainder until the remainder is zero. The last non-zero remainder is the GCD.

How do you prove the law of cosines using the geometry of a triangle?To prove the law of cosines, consider a triangle ABC with sides a, b, and c opposite angles A, B, and C, respectively. Drop a perpendicular from C to side AB, dividing it into segments of lengths x and (b-x). Using the Pythagorean theorem in the resulting right triangles, express c² in terms of a, b, and cos(C).

How can we use Bayesian statistics to analyze the probability of events when we have prior knowledge about related parameters?Bayesian statistics allows us to update the probability of an event based on prior knowledge and new data. We start with a prior distribution representing our initial beliefs about parameters. As new data becomes available, we use Bayes’ theorem to compute the posterior distribution, which combines prior beliefs and evidence. This updated distribution provides a refined probability assessment, aiding in more accurate predictions and decision-making.

How do you solve trigonometric equations involving multiple angles and identities?To solve trigonometric equations involving multiple angles and identities, first simplify the equation using trigonometric identities. Then, isolate the trigonometric function and solve for the angle. Finally, consider all possible solutions within the given range by accounting for periodicity and symmetry.

How do you find the residue of a complex function at a singular point using Laurent series expansion?To find the residue of a complex function at a singular point using Laurent series expansion, first express the function as a Laurent series around the singularity. The residue is the coefficient of the (1/(z-a)) term in this expansion, where ‘a’ is the singular point.

How do you find the sine, cosine, and tangent of an angle in a right triangle?To find the sine, cosine, and tangent of an angle in a right triangle, use the following definitions: Sine (sin) is the ratio of the length of the opposite side to the hypotenuse. Cosine (cos) is the ratio of the length of the adjacent side to the hypotenuse. Tangent (tan) is the ratio of the length of the opposite side to the adjacent side. These ratios are fundamental trigonometric functions.