How do you use the limit definition to find the slope of the tangent line to the graph # f(x) = x^2#?

Question

How do you use the limit definition to find the slope of the tangent line to the graph  # f(x) = x^2#?

Emilia Aguilar · Accepted Answer

# f'(x) = 2x # By definition # f'(x) = d/dxf(x) = lim_{h->0) (f(x+h)-f(x))/h # So if # f(x)=x^2# then: # f'(x) = lim_{h->0) ((x+h)^2-x^2)/h # # :. f'(x) = lim_{h->0) (x^2+2hx+h^2-x^2)/h # # :. f'(x) = lim_{h->0) ( 2hx+h^2 )/h # # :. f'(x) = lim_{h->0) ( 2x+h ) # # :. f'(x) = 2x #

Isaiah Allen · Answer

undefined To use the limit definition to find the slope of the tangent line to the graph of f(x) = x^2, we can follow these steps:

1. Choose a point on the graph of f(x) = x^2, let's say (a, f(a)).
2. Choose a second point on the graph that is close to (a, f(a)), let's say (a + h, f(a + h)).
3. Calculate the slope of the secant line passing through these two points using the formula: (f(a + h) - f(a)) / (a + h - a).
4. Simplify the expression: (f(a + h) - f(a)) / h.
5. Take the limit as h approaches 0 of the expression obtained in step 4.
6. The resulting limit will give us the slope of the tangent line to the graph of f(x) = x^2 at the point (a, f(a)).

By following these steps, we can find the slope of the tangent line to the graph of f(x) = x^2 using the limit definition.