To find the arc length function for the curve given by the equation y = 2x³ from the starting point P(0, 12), we will use the formula for arc length. The arc length L from point a to point b is given by the integral: L = ∫ab sqrt(1 + (dy/dx)²) dx First, we need … Read more
Finding the Vertex of a Quadratic Function The vertex of a quadratic function in the standard form f(x) = ax² + bx + c can be found using the vertex formula, which is given as: x = -b / (2a) In your case, the function is f(x) = x² + 8x + 3. Here, we … Read more
Creating Venn Diagrams for Sets A, B, and C Venn diagrams are a great way to visually represent the relationships between different sets. In this case, we’re looking at three sets: A, B, and C. Below, you’ll find detailed instructions on how to draw Venn diagrams for their various combinations, such as A ∩ B, … Read more
To find the number of units x that produces the minimum average cost per unit c given the equation c = 0.001x³ + 5x + 250, we will need to analyze the function and find its minimum point. The provided equation is a polynomial equation, and the average cost per unit is given as a … Read more
The values for the trigonometric functions at 60 degrees are: cosec 60°: The cosecant function is the reciprocal of the sine function. Therefore, cosec 60° = 1/sin(60°) = 1/(√3/2) = 2/√3 or √3/3 simplifying gives us approximately 1.1547. sec 60°: Similar to cosecant, the secant function is the reciprocal of the cosine function. So, sec … Read more
Understanding the Probability of Rolling a Sum of 9 or Higher When rolling two fair six-sided dice, we want to find the probability that the sum of the numbers on the top faces of the dice is 9 or greater. Let’s break this down step by step. Step 1: Total Possible Outcomes Each die has … Read more
The prime numbers are those natural numbers greater than 1 that can only be divided by 1 and themselves without leaving a remainder. To find the set of prime numbers that are less than 15, we can list the natural numbers starting from 2 and check each number for primality: 2 (Prime) 3 (Prime) 4 … Read more
To solve the problem, we first need to find the difference between 14 and 5. This can be calculated as follows: Difference = 14 – 5 = 9 Now we have the difference, which is 9. Next, we check the conditions given in the question: If the difference is less than 6, we add 2. … Read more
The end behavior of the logarithmic function f(x) = log3(x2) can be analyzed by considering its limits as x approaches both positive and negative infinity. First, as x approaches infinity (x → ∞), we observe that: As x becomes larger, x2 also becomes larger. Therefore, the logarithm of x2 will also increase. This means that … Read more