Math

What is the difference between mean and median in a data set and how can I calculate them?The mean is the average of a data set, calculated by summing all values and dividing by the number of values. The median is the middle value when the data set is ordered. If the number of values is even, the median is the average of the two middle numbers.

How can you solve a system of nonlinear equations using the substitution or elimination method?To solve a system of nonlinear equations using substitution, isolate one variable in one equation and substitute it into the other. For elimination, multiply equations to align terms, then add or subtract to eliminate a variable. Solve the resulting equation, then back-substitute to find the other variable.

Given sin(a + b) = sin a * cos b + cos a * sin b and sin(a – b) = sin a * cos b – cos a * sin b, derive an expression for tan(a + b) in terms of tan a and tan b.To derive tan(a + b) in terms of tan(a) and tan(b), we start with the given identities for sin(a + b) and cos(a + b). Using the identity tan(x) = sin(x)/cos(x), we get tan(a + b) = (tan(a) + tan(b)) / (1 – tan(a) * tan(b)).

What is the sine of a 30 degree angle in a right triangle?The sine of a 30-degree angle in a right triangle is 0.5. This is derived from the ratio of the length of the side opposite the angle to the length of the hypotenuse. In a 30-60-90 triangle, the sides are in the ratio 1:√3:2, making the sine of 30 degrees equal to 1/2.

If the sum of the three consecutive integers is 93, what are the integers?To find three consecutive integers whose sum is 93, let the integers be x, x+1, and x+2. Their sum is x + (x+1) + (x+2) = 93. Simplifying, we get 3x + 3 = 93. Solving for x, we find x = 30. Therefore, the three consecutive integers are 30, 31, and 32.

How do you find the equations of both the amplitude and period for a sine wave given its transformation?To find the amplitude and period of a transformed sine wave, use the general form y = A * sin(B(x – C)) + D. The amplitude is given by |A|, and the period is calculated as 2π / |B|. Here, A affects the amplitude, and B affects the period.

To find the derivative of $ \sin(x^2) $, use the chain rule:

$$ \frac{d}{dx} \sin(u) = \cos(u) \cdot \frac{du}{dx} $$

Let $ u = x^2 $, so:

$$ \frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot 2x $$

Can you explain and solve an improper integral where the integrand has an infinite discontinuity and demonstrate its convergence using the comparison test?Consider the improper integral ∫(1/x^2) dx from 1 to ∞. The integrand 1/x^2 has an infinite discontinuity at x = 0. To demonstrate convergence, compare it with ∫(1/x^2) dx from 1 to ∞, which converges because ∫(1/x^p) dx converges for p > 1. Hence, the original integral converges.

How do you solve logarithmic equations?To solve logarithmic equations, use properties of logarithms to combine or simplify terms, convert the equation to exponential form, and solve for the variable. Check for extraneous solutions by substituting back into the original equation, as logarithms are only defined for positive arguments.

How do you simplify (x^(1/3) + x^(-1/3))^2?