How do you use the limit definition to find the slope of the tangent line to the graph #f(x)=x^3#?

Question

How do you use the limit definition to find the slope of the tangent line to the graph  #f(x)=x^3#?

William Abdalla · Accepted Answer

Expand, reduce and evaluate the limit.

Definition 1 If you are using as your definition of the slope at #x#: #lim_(hrarr0)(f(x+h)-f(x))/h#, then you'll need to expand #(x+h)^3 = x^3+3x^2h+3xh^2+h^3#. (If using the slope at #a# rather than at #x#, simply replace the #x#'s above with #a#'s.) Definition 2 If you are using the slope at #a# defined by #lim_(xrarra)(f(x)-f(a))/(x-a)#, then you'll need to factor #x^3-a^3 = (x-a)(x^2+2ax+a^2)# (difference of cubes).

Scarlett Allison · Answer

undefined To use the limit definition to find the slope of the tangent line to the graph of f(x) = x^3, we can follow these steps:

1. Start with the equation of the function: f(x) = x^3.

2. Choose a point on the graph of the function, let's say (a, f(a)).

3. Select a second point on the graph that is very close to the first point, let's say (a + h, f(a + h)).

4. Calculate the slope of the secant line passing through these two points using the formula: (f(a + h) - f(a)) / (a + h - a).

5. Simplify the expression obtained in step 4.

6. Take the limit as h approaches 0 of the expression obtained in step 5.

7. The result of step 6 will give us the slope of the tangent line to the graph of f(x) = x^3 at the point (a, f(a)).

Therefore, by following these steps, we can use the limit definition to find the slope of the tangent line to the graph of f(x) = x^3.