Use the four-step process to find the slope of the tangent line to the graph of the given function at any point.
f(x) = 16 - 10 x
Step 2: f(x + h) - f(x) =
Step 4: f'(x) = lim(h->0)(f(x + h)- f(x))/h =
-10(x+h) + 16 -(-10x +16)/h
(-10x -10h +16 + 10x -16)/h
-10h/h
lim(h->0) = -10
Step 2: To find the slope of the tangent line to the graph of the function at any point, we need to find the difference quotient, which represents the average rate of change of the function over a small interval.
To find f(x + h), we substitute (x + h) into the function:
f(x + h) = 16 - 10(x + h) = 16 - 10x - 10h
Subtracting f(x) from f(x + h):
f(x + h) - f(x) = (16 - 10x - 10h) - (16 - 10x)
= -10h
Step 4: To find the slope of the tangent line, we use the definition of the derivative.
f'(x) = lim(h->0)(f(x + h) - f(x))/h
Substituting the difference quotient:
f'(x) = lim(h->0)(-10h)/h
Simplifying:
f'(x) = lim(h->0)-10
Taking the limit as h approaches 0, the slope of the tangent line is -10.
Step 2: To find the slope of the tangent line, we need to calculate the difference quotient. Let's plug in the values into the formula:
f(x + h) - f(x) = (16 - 10(x + h)) - (16 - 10x)
Simplifying further:
f(x + h) - f(x) = 16 - 10x - 10h - 16 + 10x
This simplifies to:
f(x + h) - f(x) = -10h
Step 4: Now, to find the derivative (slope of the tangent line), we need to take the limit as h approaches 0 of the difference quotient:
f'(x) = lim(h->0) (-10h / h)
Simplifying further:
f'(x) = lim(h->0) -10
Therefore, the slope of the tangent line to the graph of the function f(x) = 16 - 10x at any point is -10.