What is the integration of x?
The integration of x with respect to x is a fundamental concept in calculus. The integral of x is expressed mathematically as: ∫ x dx = (1/2)x² + C Here, C represents the constant of integration, which is included because integration can yield a family of functions that differ by a constant. To understand why […]
The measure of central angle RST is π radians. What is the area of the shaded sector?
To find the area of the shaded sector, we can use the formula for the area of a sector of a circle: Area of Sector = (θ/2) * r² where θ is the central angle in radians, and r is the radius of the circle. In this case, we know that the central angle RST […]
What is the equation of a circle with center (3, 2) and radius 5?
The equation of a circle in the Cartesian coordinate system can be formulated using the standard equation: (x – h)² + (y – k)² = r² In this equation, (h, k) represents the center of the circle and r represents the radius. Given the center (3, 2) and radius 5, we can substitute these values […]
Does every polynomial function of degree 3 with real coefficients have exactly three real zeros?
No, every polynomial function of degree 3 with real coefficients does not necessarily have exactly three real zeros. In fact, a cubic polynomial can have one real zero and two complex zeros. To understand this better, consider the general form of a cubic polynomial: f(x) = ax³ + bx² + cx + d, where a, […]
What is the range of the relation R = {(3, 5), (1, 2), (1, 4), (1, 2)}?
The range of a relation is the set of all second elements (outputs) from the ordered pairs in that relation. In the given relation R, we can look at each ordered pair: (3, 5) – Here, the second element is 5. (1, 2) – The second element is 2. (1, 4) – The second element […]
What number should be added to both sides of the equation to complete the square x² + 6x – 5?
To complete the square for the quadratic expression x² + 6x – 5, we need to turn the expression into a perfect square trinomial. The first step is to take the coefficient of the linear term (which is 6 in this case), divide it by 2, and then square the result. So, we compute: Divide […]
What is the derivative of f(x) = tan⁻¹(x)?
The derivative of the function f(x) = tan⁻¹(x) is given by: f'(x) = 1 / (1 + x²) To understand why this is the case, let’s look at the process of differentiation for the inverse tangent function. The function tan⁻¹(x) represents the angle whose tangent is x. When we differentiate this using implicit differentiation, we […]
How large is the central angle on a circle with a radius of 4ft if the defined arc is 10ft?
To find the central angle of a circle based on the length of the arc, we can use the relationship between the arc length, the radius, and the central angle in radians. The formula for arc length (L) is given by: L = r * θ Where: L is the length of the arc r […]
How do you solve the equation 3x + 6x = 1?
To solve the equation 3x + 6x = 1, you first combine like terms. The terms 3x and 6x can be added together, which gives: 3x + 6x = 9x This simplifies the equation to: 9x = 1 The next step is to isolate x. You can do this by dividing both sides of the […]
If f(x) = x^2 + 1 and g(x) = x + 4, which value is equivalent to f(g(10))?
To find the value of f(g(10)), we need to first calculate g(10) and then use that result to find f. Starting with g(x): g(x) = x + 4 Plugging in 10: g(10) = 10 + 4 = 14 Now that we have g(10), we can find f(g(10)) or f(14): f(x) = x² + 1 So, […]