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The Mean Value Theorem (MVT) states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in (a, b) such that: f'(c) = (f(b) – f(a)) / (b – a) To apply the MVT to the function … Read more

To determine the probability of the union of two mutually exclusive events A and B, we can use the formula: P(A ∪ B) = P(A) + P(B) In our case, we have: P(A) = 0.2 P(B) = 0.6 Since events A and B are mutually exclusive, this means that they cannot happen at the same … Read more

To find the probability that he knew the answer to the question given that he answered it correctly, we can employ Bayes’ theorem. Bayesian probability allows us to update our beliefs based on new evidence. In this case, we’re interested in the conditional probability. Let’s define the following events: K: He knew the answer. C: … Read more

To solve the quadratic equation x² – 6x – 22 = 0, we can use the quadratic formula, which is given by: x = (-b ± √(b² – 4ac)) / (2a) In this equation: a = 1 b = -6 c = -22 First, we need to calculate the value of the discriminant, b² – … Read more

To find the sum of the numbers 3, 1421, 0, 241, 0, and 9, we will simply add them together. Let’s break it down: 3 + 1421 = 1424 1424 + 0 = 1424 1424 + 241 = 1665 1665 + 0 = 1665 1665 + 9 = 1674 Now, the final sum is 1674. … Read more

To analyze the question, we need to first clarify what it means for a function to produce a specific output for a given input. In this case, we have a function f such that when the input is 1, the output is 5. This means that f(1) = 5. However, we must consider other possible … Read more

To find the completely factored form of the expression x4y + 4x2y + 5y, we can start by factoring out the common terms. 1. Identify Common Factors: The expression has a common factor of y. 2. Factor Out ‘y’: By factoring out ‘y’, we get: y(x4 + 4×2 + 5) 3. Next, we need to … Read more

To find the exact length of the curve defined by the equation y = 18×3 + 20x + 1, we use the formula for the arc length of a curve. The arc length L of a function y = f(x) from x = a to x = b is given by: L = ∫ab √(1 … Read more

To find the number of two-digit positive integers where the product of their digits equals 24, we start by identifying the two digits of a two-digit number, which we can denote as AB, where A is the tens digit and B is the units digit. Given that the product of the digits is stated as: … Read more

To express the complex number given as 6cos(330°) + i sin(330°) in the standard form a + bi, we first need to calculate the values of cos(330°) and sin(330°). Step 1: Calculate cos(330°) The angle 330° is in the fourth quadrant of the unit circle. In this quadrant, cosine values are positive. The reference angle … Read more