How Do You Find the Area of a Circle with the Diameter?
To find the area of a circle when you have the diameter, you first need to remember the formula for the area of a circle, which is: Area = πr² In this formula, r represents the radius of the circle. The radius is half of the diameter, so if you know the diameter (d), you […]
If cx = 5x^2 and dx = x^3, what is the domain of cdx?
To find the domain of the product cdx
How do I graph the solution sets of the inequalities xy ≤ 4?
To graph the solution set of the inequality xy ≤ 4, we start by rewriting it in a more usable form. We can express it as: y ≤ 4/x Next, we need to consider the equation y = 4/x. This represents a hyperbola. To graph the inequality, follow these steps: Identify the intercepts: The curve […]
What is the derivative of log x?
The derivative of log x, where log denotes the natural logarithm (ln), is given by: f'(x) = 1/x To understand why this is the case, let’s recall the definition of the natural logarithm. The natural logarithm is the inverse function of the exponential function. That is, if y = ln(x), then x = ey. To […]
Which equation has only one solution?
An equation that has only one solution is generally referred to as a linear equation. A common form of a linear equation is ax + b = c, where a, b, and c are constants and a is not equal to zero. When we solve for x, we find a unique value that satisfies the […]
What does the integral tan(x) sin(x) sin(y) dx cos(x) cos(y) dy evaluate to when set to 0?
To evaluate the expression tan(x) sin(x) sin(y) dx cos(x) cos(y) dy and determine when it’s equal to zero, we first need to understand the individual components involved in the expression. The expression consists of: tan(x): The tangent function, which can be expressed as sin(x)/cos(x). sin(x) sin(y): The product of sine functions, which vary between -1 […]
Express the series in summation notation: 2, 4, 6, 8, 10, 12
The given series is: 2, 4, 6, 8, 10, 12. This is an arithmetic series where each term increases by 2. To express this series in summation notation, we first identify a general term for the series. We can observe that the nth term can be expressed as: an = 2n for n = 1, […]
What is the solution set of 18 < 3n < 20 < 4n?
To solve the double inequality 18 < 3n < 20 < 4n, we need to break it down into two separate inequalities and solve them one at a time. 1. First, let’s solve the part 18 < 3n: Divide both sides by 3: 6 < n 2. Now, let’s solve the part 20 < 4n: […]
How to Solve the Differential Equation 1 4y” + 3y’ = x^2?
To solve the differential equation 1 4y” + 3y’ = x^2 by the method of undetermined coefficients, we first rewrite it in standard form: 4y” + 3y’ = x^2 Next, we need to find the complementary solution (the solution to the homogeneous equation) and the particular solution (the solution to the non-homogeneous equation). Step 1: […]
What is the value of y in the solution to the system of equations x + y = 1 and 2x + 3y = 30?
To find the value of y in the system of equations: x + y = 1 2x + 3y = 30 We can solve this system using the substitution or elimination method. Here, I’ll use the substitution method for clarity: First, let’s solve the first equation for x: x = 1 – y Next, we […]