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The graph of y = sin(x) is a periodic wave that oscillates between -1 and 1. To understand where it starts for the specified domain from x = 0 to x = 2π, let’s look at the behavior of the sine function at these crucial points. When x = 0: The sine of 0 is … Read more

To evaluate the limit by recognizing it as a Riemann sum for a function defined on the interval [0,1], we first need to understand what a Riemann sum represents. A Riemann sum approximates the integral of a function over a specified interval by dividing that interval into smaller subintervals. The limit in question often appears … Read more

Understanding Polynomial Division In polynomial division, we divide one polynomial by another, similar to how we would do with numbers. Here, we want to find the quotient of the polynomial x³ + 3x² + 3x + 2 by the polynomial x² + x + 1. Step 1: Set Up the Division We begin by setting … Read more

The function f(x) = tan(4x) has vertical asymptotes wherever the tangent function is undefined. This occurs at odd multiples of π/2. To find the specific asymptotes for the function over the interval from x = 0 to x = 2, we first need to set the equation: 4x = (2n + 1)π/2 where n is … Read more

To solve the system of equations 2y = 6 and 2x + 4y = 10, we can use substitution or elimination methods. Here, we’ll outline the substitution method. Step 1: Solve for y First, simplify the first equation to find the value of y: 2y = 6 => y = 6 / 2 => y … Read more

In a geometric series, we denote the first term as a1 and the common ratio as r. Let’s analyze the series given: 2, 2, 2, 2, 2. First term (a1): The first term of this geometric series is simply the first number listed in the series, which is 2. Therefore, a1 = 2. Common ratio … Read more

In the quadratic equation of the standard form, which is typically given as y = ax2 + bx + c, the coefficients a, b, and c are critical for defining the shape and position of the parabola on a graph. For the specific quadratic equation: y = 5×2 + 4x + 2 We can identify … Read more

To find f(3) for the function f(x) = 2a + 3x + 9ax, we will substitute x = 3 into the function. 1. Start with the function: f(x) = 2a + 3x + 9ax 2. Substitute x with 3: f(3) = 2a + 3(3) + 9a(3) 3. Now calculate the individual terms: 3(3) = 9 … Read more

To express the number 30 as a fraction in its lowest terms, we first need to understand what it means to write a number as a fraction. A fraction consists of a numerator and a denominator. In this case, we can represent 30 as: 30/1 This fraction is in its simplest form because the numerator … Read more

To find the domain of the composite function fg(x), we first need to determine the domains of the individual functions f(x) and g(x). Step 1: Determine the domain of g(x) Given g(x) = x + 5, this is a linear function. The domain of g(x) is all real numbers, represented as: Domain of g(x): (-∞, … Read more