Question - Derivative Calculation with a Function and Point on a Graph
Solution:
The image includes a graph displaying a straight line representing a function y = f(x), along with a pair of points labeled A(2, 1). Additionally, there's a mathematical question that states:Assuming the function g(x + 1) = (x^2 + 1) - f(x), what is the value of g'(3)?First, let's determine the slope of the line representing y = f(x) since we have the point A(2, 1) on the line. Since the line passes through the point A(2, 1) and the origin (0, 0), we can use these two points to calculate the slope (m):m = (y2 - y1) / (x2 - x1)m = (1 - 0) / (2 - 0)m = 1 / 2Thus, the slope of the line is 1/2, and knowing that a line equation with slope m passing through the origin (0,0) has the form y = mx, the equation for f(x) is:f(x) = (1/2)xNow, let's put f(x) into the equation for g(x + 1):g(x + 1) = (x^2 + 1) - (1/2)xTo find g'(x), the derivative of g(x), we must first write g(x) in terms of x:g(x) = ((x - 1)^2 + 1) - (1/2)(x - 1)g(x) = (x^2 - 2x + 1 + 1) - (1/2)x + (1/2)g(x) = x^2 - (5/2)x + 2The next step is to differentiate g(x) with respect to x to find g'(x):g'(x) = 2x - 5/2Finally, to find g'(3), substitute 3 into the derivative:g'(3) = 2(3) - 5/2g'(3) = 6 - 5/2g'(3) = (12/2) - (5/2)g'(3) = (12 - 5)/2g'(3) = 7/2Therefore, g'(3) is 7/2.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com