How are these calculus questions different?
How are a) and b) different? What should be done for each one?
- Let f(x) = x^2 - 4x - 5
a) Find f '(3) using the definition f a = lim h->0 f(a+h) - f(a)/h
b) Find the slope of the tangent line to the graph of f(x) at x=3
(I thought slope of the tangent line at x=3 was the same as finding the derivative of 3?)
How are a) and b) different? What should be done for each one?
- Let f(x) = x^2 - 4x - 5
a) Find f '(3) using the definition f a = lim h->0 f(a+h) - f(a)/h
b) Find the slope of the tangent line to the graph of f(x) at x=3
(I thought slope of the tangent line at x=3 was the same as finding the derivative of 3?)
2 Answers
Both answers are 2.
Explanation:
In the case of question (a), you are being asked to find the derivative
In this case, the limit definition essentially describes the concept of using secant lines to approximate the slope of a tangent to a function, while gradually reducing the "spacing" between the two
In the graph, the solid green curve is
The slope is traditionally phrased as "rise over run", meaning the amount the function changes in the y-direction (
Likewise, we know the difference between the two x coordinates is simply
This gives us an expression for the slope of the line connecting
The "magic" comes from then using a limit to mathematically "see" what happens when we make the separation
What this limit says is if you let the two points get closer and closer together, the slope will approach the value 2. (If you are patient, you can see that in my picture above, as the slopes of the dotted lines are 2.75, 2.5, and 2.25 - all values heading towards 2.)
In question (b), you are correct in noting that the slope of the tangent line to
Check your textbook/teacher's definitions.
Explanation:
Sometimes (for example in James Stewart's text) the slope of the tangent line at
#m = lim_(xrarra)(f(x)-f(a))/(x-a)# if the limit exists.
While the derivative of
#f'(a) = lim_(hrarr0)(f(a+h)-f(a))/h# if the limit exists.
Theorem: The two limits above either both exist and are equal or neither exists.
So the numerical answer will be the same either way.