Find the Length of the Curve r(t) = rt cos(2t) i + sin(2t) j + 2 ln(cos(t)) k, 0 ≤ t ≤ π/4
To find the length of the curve defined by the vector function r(t) = (r(t) cos(2t), sin(2t), 2 ln(cos(t))) over the interval [0, π/4], we need to use the formula for the arc length of a parametric curve. The formula for the length L of a curve given by a vector function r(t) from t […]
What is the range of the function f(x) = 3x² + 6x + 8?
To find the range of the function f(x) = 3x² + 6x + 8, we first need to analyze its shape and vertex, as it is a quadratic function. The general form of a quadratic function is f(x) = ax² + bx + c. For our function, a = 3, b = 6, and c […]
How do you simplify the rational expression 10n^2 / (24n^4 + 9n^2 + 18) and what are the restrictions on the variable?
To simplify the rational expression 10n2 / (24n4 + 9n2 + 18), we first need to factor the denominator. The expression in the denominator is 24n4 + 9n2 + 18. We can factor this expression by looking for common factors and simplifying it step-by-step: First, we can factor out a common factor of 3: 3(8n4 […]
Find the Rectangular Coordinates of the Point with the Polar Coordinates 3 and 3π/2
To convert polar coordinates to rectangular coordinates, we can use the formulas: x = r * cos(θ) y = r * sin(θ) Here, the polar coordinates given are (r, θ) = (3, 3π/2). In this case, r = 3 and θ = 3π/2 radians. Now, we calculate the x and y coordinates: x = 3 […]
In the diagram of circle C, what is the measure of 1?
To determine the measure of the angle or segment labeled ‘1’ in the diagram of circle C, we first need to analyze the characteristics of the circle and the positions of the points and lines involved. If ‘1’ refers to an angle formed by two radii or a central angle, its measure can typically be […]
What is the slope of the line passing through the points (2, 5) and (4, 1)?
To find the slope of the line passing through the two points, we can use the formula for slope, which is: slope (m) = (y2 – y1) / (x2 – x1) Here, we have two points: (x1, y1) = (2, 5) and (x2, y2) = (4, 1). Plugging the coordinates into the formula, we get: […]
What Do We Mean by Square and Triangle Numbers?
Square numbers and triangle numbers are two special categories of numbers that are often studied in mathematics. Square Numbers: A square number is an integer that can be expressed as the product of another integer multiplied by itself. In simpler terms, it’s the result of squaring a whole number. For example, the first few square […]
Determine the Type of Boundary Line and Shading for the Graph of the Inequality 2x + y < 4
To analyze the inequality 2x + y < 4, we first need to convert it into an equation to find out the boundary line. The equation corresponding to the inequality is: 2x + y = 4 Next, we determine the type of boundary line. Since the inequality is a strict one (
What is the quotient of x^3 + 3x^2 + 5x + 3?
To find the quotient of the polynomial x3 + 3×2 + 5x + 3, we can use polynomial long division to divide it by a linear term, if that’s what you meant. However, your question seems to only present a single polynomial, so it’s a bit unclear what you’re dividing by. If we assume you […]
Evaluate lim h → 0 √(9h) / (3h)
To evaluate the limit lim h → 0 √(9h) / (3h), we can start by simplifying the expression. First, we note that the square root term can be simplified: √(9h) can be rewritten as √9 * √h = 3√h. Now, we can rewrite the limit: lim h → 0 (3√h) / (3h) = lim h […]