The function f(x) = x² represents a parabolic graph that opens upwards. When comparing the graphs of f(x) = x³ and f(x) = x², we notice distinct characteristics due to their different degrees and forms. 1. Shape: The graph of f(x) = x² is a smooth, U-shaped curve, known as a parabola. This contrasts sharply … Read more
To analyze the system of equations given by y = 2×2 and y = 2x, we start by setting both expressions for y equal to each other: 2×2 = 2x Next, we can simplify this equation by dividing both sides by 2 (assuming x is not zero): x2 = x Rearranging this gives us: x2 … Read more
The mean, commonly known as the average, plays a significant role in various aspects of our daily routine, often without us even realizing it. Here are several areas where we encounter the mean: Finance: One of the most common places we use the mean is in managing personal finances. For instance, when evaluating how much … Read more
To evaluate the integral ∫ x³ cos(x²) dx, we will first use substitution and then apply integration by parts. Step 1: Substitution We start with a substitution that simplifies our integral. Let’s set: u = x² Then, the differential du is: du = 2x dx or dx = rac{du}{2x} Now, we can express x³ in … Read more
To determine which positive integers less than 12 are relatively prime to 12, we first need to understand what ‘relatively prime’ means. Two numbers are considered relatively prime (or coprime) if their greatest common divisor (GCD) is 1. In other words, they do not share any prime factors. The positive integers less than 12 are: … Read more
If f(5) = 0, this indicates that x = 5 is a root of the polynomial function f(x). According to the Remainder Theorem, if you divide a polynomial f(x) by (x-c), the remainder of this division is equal to f(c). Since f(5) = 0, it means that (x – 5) is a factor of the … Read more
To find the value of m when m = 3 is a solution of the equation 4x + 9y = 3, we can substitute x and y values into the equation. In this case, let’s assume that x = m and y = 3. This gives us: 4(m) + 9(3) = 3 Now, we can … Read more
To determine if the set of positive integers (denoted as ℕ+ ) forms a group under the operation of addition, we need to evaluate it against the four fundamental properties that define a mathematical group: Closure: A set is closed under an operation if performing that operation on members of the set results in another member … Read more
To find the slope of the curve at a specific point and the equation of the tangent line, follow these steps: Step 1: Identify the function and the point The given function is: y = 6×2 – 19 We need to find the slope at point P. Let’s assume point P is (p, y). To … Read more
To determine the value of c that makes the equation 2x3c × x3 = x2 + 10x + 6 + 2×5 true, we first need to equate both sides of the equation and simplify. Start by simplifying the left-hand side. Left-hand side: 2x3c × x3 = 2c × x6 Now the left-hand side can be … Read more