Math

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.

How do you find the transformations applied to the graph of a function f(x) represented by f(x) = -2(x – 4)² + 3?To find the transformations applied to the graph of f(x) = -2(x – 4)² + 3, identify the following: horizontal shift 4 units right (x – 4), vertical stretch by a factor of 2, reflection across the x-axis (-2), and vertical shift 3 units up (+3).

How can we determine whether a time series data is stationary and which tests can be used to check for stationarity?To determine if a time series is stationary, we check if its statistical properties like mean, variance, and autocorrelation are constant over time. Common tests for stationarity include the Augmented Dickey-Fuller (ADF) test, the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test, and the Phillips-Perron (PP) test.

What are the different types of angles and how are they measured?Angles are classified based on their measures. Acute angles are less than 90 degrees, right angles are exactly 90 degrees, obtuse angles are between 90 and 180 degrees, straight angles are exactly 180 degrees, and reflex angles are between 180 and 360 degrees. They are measured using a protractor.

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).

What are the basic steps for adding and subtracting fractions with different denominators?To add or subtract fractions with different denominators, follow these steps: 1) Find the least common denominator (LCD) of the fractions. 2) Convert each fraction to an equivalent fraction with the LCD. 3) Add or subtract the numerators, keeping the denominator the same. 4) Simplify the resulting fraction if possible.