How to Find the Linearization of the Function at a Given Point?
To find the linearization, or the linear approximation, of the function at a point, we follow a few simple steps. Let’s consider the function f(x) and the point where we want to linearize it. 1. **Identify the function and point**: First, we need to know the function f(x) and the point a at which we […]
What is the value of sin(8) given that (3, 4) is a point on the terminal side of 8?
To find the value of sin(8) using the point (3, 4) on its terminal side, we first need to determine the hypotenuse of the right triangle formed by this point. Using the Pythagorean theorem, we can calculate the hypotenuse (r): r = √(x² + y²) Substituting the coordinates: r = √(3² + 4²) = √(9 […]
What are the x and y intercepts of the graph of y = x^2 + 10x + 21?
To find the x-intercepts of the graph of the equation y = x2 + 10x + 21, we need to set y to zero and solve for x. This gives us: x2 + 10x + 21 = 0 We can factor this quadratic equation. (x + 3)(x + 7) = 0 Setting each factor to […]
Given that P(2, 9) and Q(4, 14), find the component form and magnitude of the vector PQ
To find the component form of the vector PQ, we first need to determine the coordinates of points P and Q. Point P has coordinates (2, 9) and point Q has coordinates (4, 14). The component form of a vector PQ can be expressed as: PQ = Q – P Calculating the components: For the […]
Find the Equation for a Parabola with its Focus at (0, 3) and a Directrix of y = 3
To find the equation of a parabola given its focus and directrix, we start by understanding their definitions. The focus of a parabola is a fixed point used to define the curve, while the directrix is a line. The parabola is defined as the set of all points that are equidistant from the focus and […]
Which of the following equations has only one solution?
To determine which equation has only one solution, we need to examine the given options. An equation has only one solution if it can be expressed in the form of a perfect square trinomial, where the discriminant (b² – 4ac) is equal to zero. Let’s analyze the equations: x² – 9 = 0: This can […]
How do you find unit vectors that are orthogonal to both i, j and i, k?
To find a unit vector that is orthogonal to both vectors i and j (which represent the x-axis and y-axis respectively in 3D space), as well as to both i and k (which represent the x-axis and z-axis), we can proceed as follows: First, we can identify the vectors: i = (1, 0, 0) j […]
How do you find the linearization L(x) of the function f(x) at a = 4?
To find the linearization L(x) of the function f(x) at a specific point, we first need to calculate the derivative of the function and the value of the function itself at that point. Here’s the general formula for the linearization of a function f(x) around a point a: L(x) = f(a) + f'(a)(x – a) […]
Find the Circumference of a Circle if the Area is 300 Square Centimeters
To find the circumference of a circle when you know its area, you can start by using the formula for the area of a circle, which is: Area = πr² In this case, the area is given as 300 square centimeters: 300 = πr² To find the radius (r), we will rearrange this formula: r² […]
Is the inequality 22x + 9 < 4x + 9 sometimes, always, or never true?
To determine whether the inequality 22x + 9 < 4x + 9 is sometimes, always, or never true, we can start by simplifying the inequality. First, we can subtract 4x from both sides: 22x – 4x + 9 < 9 This simplifies to: 18x + 9 < 9 Next, let’s subtract 9 from both sides: […]