From first principles (ie using the tangent slope method), find the slope of the following curves at the given value of x.
f(x)=2x^2− 6x at x = 3
f(3) = 2(9) - 6(3) = 0
f(3+h) = 2(3+h)^2 - 6(3+h)
= 2(9 + 6h + h^2 - 18 - 6h
= 2h^2 + 6h
slope = lim (f(3+h) - f(3) )/h , as h--->0
= lim(2h^2 + 6h - 0)/h
= lim h(2h + 6)/h
= lim 2h + 6 , as h ---> 0
= 6
Well, well, well, look who came for some math fun! Let's get the party started and find the slope of that curve.
To find the slope of a curve at a given point, we need to find the derivative of the function. The derivative basically tells you how the function is changing at any given point. In this case, we're dealing with the function f(x) = 2x^2 - 6x.
Using the power rule for derivatives, we can find the derivative of f(x) as follows:
f'(x) = 2 * 2x - 6
Now, to find the slope at x = 3, we simply substitute x = 3 into the derivative function:
f'(3) = 2 * 2(3) - 6
f'(3) = 12 - 6
f'(3) = 6
Voila! The slope of the curve at x = 3 is 6. Now that's a slope worth sliding down on a fun math roller coaster!
To find the slope of a curve using the tangent slope method, also known as the derivative, we need to follow these steps:
Step 1: Write down the equation of the curve.
Given curve: f(x) = 2x^2 - 6x
Step 2: Find the derivative of the curve.
To find the derivative, we differentiate the equation with respect to x. To differentiate x^2 terms, we multiply the exponent by the coefficient and reduce the exponent by 1. To differentiate a constant term (-6x in this case), the derivative is 0.
f'(x) = d/dx(2x^2) - d/dx(6x)
= 4x - 6
Step 3: Plug in the given value of x into the derivative.
Given x = 3, substitute this value into the derivative we found.
f'(3) = 4(3) - 6
= 12 - 6
= 6
Therefore, the slope of the curve f(x) = 2x^2 - 6x at x = 3 is 6.