2.3 Differentiabilityap Calculus
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Differentiable means that the derivativeexists ..
Lesson 2.6: Diﬀerentiability: Afunctionisdiﬀerentiable at a point if it has a derivative there. In other words: The function f is diﬀerentiable at x if.
Posted by 3 years ago. Can someone help me with this question? (Ap Calc AB - Graphing/differentiability) Ap calculus. Differentiate any function with our calculus solver. Enter an expression and the variable to differentiate with respect to. Then click the Differentiate button. FREE-RESPONSE SOLUTIONS 2019 AB Question AB-2 (a) vtP is differentiable and therefore continuous on 0.3,2.8.Since 2.8 0.3 55 55 0 2.8 0.3 2.5 vvPP. Its derivative is (1/3)x −(2/3) (by the Power Rule) At x=0 the derivative is undefined, so x (1/3) is not differentiable. At x=0 the function is not defined so it makes no sense to ask if they are differentiable there.
Example: is x^{2} + 6x differentiable?
Derivative rules tell us the derivative of x^{2} is 2x and the derivative of x is 1, so:
Its derivative is 2x + 6
So yes! x^{2} + 6x is differentiable.
.. and it must exist for every value in the function's domain.
DomainIn its simplest form the domain is |
Example (continued)
When not stated we assume that the domain is the Real Numbers.
For x^{2} + 6x, its derivative of 2x + 6 exists for all Real Numbers.
So we are still safe: x^{2} + 6x is differentiable.
But what about this:
Example: The function f(x) = x (absolute value):
x looks like this: |
At x=0 it has a very pointy change!
Does the derivative exist at x=0?Format external hard drive for mac using windowslasopafs.
Testing
We can test any value 'c' by finding if the limit exists:
limh→0f(c+h) − f(c)h
Example (continued)
Let's calculate the limit for x at the value 0:
The limit does not exist! To see why, let's compare left and right side limits:
The limits are different on either side, so the limit does not exist.
So the function f(x) = x is not differentiable
A good way to picture this in your mind is to think:
As I zoom in, does the function tend to become a straight line?
The absolute value function stays pointy even when zoomed in.
Other Reasons
Here are a few more examples:
The Floor and Ceiling Functions are not differentiable at integer values, as there is a discontinuity at each jump. But they are differentiable elsewhere. |
The Cube root function x^{(1/3)} Its derivative is (1/3)x^{−(2/3)} (by the Power Rule) At x=0 the derivative is undefined, so x^{(1/3)} is not differentiable. |
At x=0 the function is not defined so it makes no sense to ask if they are differentiable there. To be differentiable at a certain point, the function must first of all be defined there! |
As we head towards x = 0 the function moves up and down faster and faster, so we cannot find a value it is 'heading towards'. So it is not differentiable. |
Different Domain
But we can change the domain!
Example: The function g(x) = x with Domain (0,+∞)
The domain is from but not including 0 onwards (all positive values).
2.3 Differentiabilityap Calculus Test
Which IS differentiable.
And I am 'absolutely positive' about that :)
So the function g(x) = x with Domain (0,+∞) is differentiable.
We could also restrict the domain in other ways to avoid x=0 (such as all negative Real Numbers, all non-zero Real Numbers, etc).
Why Bother?
Because when a function is differentiable we can use all the power of calculus when working with it.
Continuous
When a function is differentiable it is also continuous.
Differentiable ⇒ Continuous
But a function can be continuous but not differentiable. For example the absolute value function is actually continuous (though not differentiable) at x=0.
(1) Find the derivatives of the following functions using first principle.
(i) f(x) = 6Solution
(ii) f(x) = -4x + 7Solution
(iii) f(x) = -x^{2} + 2 Solution
(2) Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?
(i) f(x) = x - 1 Solution
(ii) f(x) = √(1 - x^{2}) Solution
(3) Determine whether the following function is differentiable at the indicated values.
(i) f(x) = x x at x = 0 Solution
(ii) f(x) = x^{2} - 1 at x = 1 Solution
(iii) f(x) = x + x - 1 at x = 0, 1 Solution
(iv) f(x) = sin x at x = 0 Solution
(4) Show that the following functions are not differentiable at the indicated value of x.
(i)
Solution
(5) The graph of f is shown below. State with reasons that x values (the numbers), at which f is not differentiable.
Solution
(6) If f(x) = x + 100 + x^{2}, test whether f'(-100) exists.
(7) Examine the differentiability of functions in R by drawing the diagrams.
(i) sin x Solution
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(ii) cos x Solution
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