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To determine the value of k given the functions f(x) and g(x), you’ll need to carefully analyze the graphs of these functions. Here’s a step-by-step approach to guide you through the process: Plot the Functions: Start by plotting both f(x) and g(x) on the same graph. Make sure to label the axes clearly and include … Read more

To find the equation of the line in slope-intercept form (y = mx + b) that passes through the points (6, 1) and (3, 2), we will follow these steps: Step 1: Calculate the slope (m) The slope (m) of a line can be found using the formula: m = (y2 – y1) / (x2 … Read more

Finding the Third-Degree Taylor Polynomial for f(x) = tan^{-1}(x) Centered at x = 7 The Taylor polynomial of a function f centered at a point a is given by the formula: T_n(x) = f(a) + f'(a)(x – a) + (f”(a)/2!)(x – a)^2 + (f”'(a)/3!)(x – a)^3 + … + (f^{(n)}(a)/n!)(x – a)^n In our case, … Read more

To graph the composite function h(x) = f(g(x)) using the graphs of f and g, you’ll follow a step-by-step approach that ensures you’re capturing the necessary transformations accurately. Here’s how to do it: Step 1: Understand your functions Make sure you have the graphs of both functions f(x) and g(x). Identify the segments of these … Read more

Understanding Direct Variation When we say that y varies directly with x, it means that the ratio of y to x remains constant. This implies that we can express this relationship as: y = kx where k is a constant of variation. To find the value of k, we can use the values provided: y … Read more

To determine the value of the constant c that makes the function f continuous, we need to analyze the function at the points where the definition changes. The function is defined as: f(x) = cx2 + 2x + 1 for x < 2 f(x) = cx for x > 2 The function will be continuous … Read more

To simplify the expression 6×4 + 16×2 + 9x + 1, we will look for common factors and see if we can factor it further. First, let’s rewrite the expression clearly: 6×4 + 16×2 + 9x + 1 Now, notice that this expression doesn’t have a straightforward factorization due to the different powers of x. … Read more

To find two pairs of polar coordinates for the point (3, 3), we first need to understand the relationship between Cartesian coordinates and polar coordinates. In polar coordinates, a point is defined by a radius (r) and an angle (θ). The conversion from Cartesian to polar coordinates uses the formulas: r = √(x² + y²) … Read more

To find dy/dx using implicit differentiation for the equation x² + 8xy + y² = 8, follow these steps: Start with the given equation: x² + 8xy + y² = 8 Differentiate both sides with respect to x. Remember, when you differentiate terms involving y, you will need to apply the chain rule and include … Read more

An arithmetic sequence is defined as a sequence of numbers in which the difference between consecutive terms is constant. To find the nth term of an arithmetic sequence, we typically use the formula: a(n) = a(1) + (n – 1) * d Where: a(n) is the nth term of the sequence. a(1) is the first … Read more