Use the limit process to find the slope of the graph of
sqrt(x + 8) at (8, 4).
Just so you can check your work, you should come up with 1/8
slope= lim (sqrt(x+8+dx)-sqrt(x+8))/dx
multipy num/den by sqrt(x+8+dx)+sqrt(x+8)
slope= lim (x+8+dx-x-8)/(dx*(sqrt(x+8+dx)+sqrt(x+8)
slope= lim ((dx/dx)/sqrt(x+8+dx)+sqrt(x+8)
= 1/(2(sqrt(x+8))
so at x=8, slope= 1/8
To find the slope of the graph of sqrt(x + 8) at the point (8, 4) using the limit process, we can follow these steps:
Step 1: Write the equation of the given function: f(x) = sqrt(x + 8).
Step 2: Calculate the derivative of the function f'(x) using the rules of differentiation. In this case, since we have a square root function, we can use the power rule.
f'(x) = (1/2)*(x + 8)^(-1/2)
Step 3: Substitute the x-coordinate of the given point (8, 4) into the derivative function to evaluate the derivative at that point.
f'(8) = (1/2)*(8 + 8)^(-1/2) = (1/2)*16^(-1/2) = (1/2)*(1/4) = 1/8
Step 4: The slope of the graph at the point (8, 4) represents the instantaneous rate of change of the function at that point. Therefore, the slope of the graph of sqrt(x + 8) at (8, 4) is 1/8.
Using the limit process to find the slope involves calculating the derivative of the function and evaluating it at the desired point.