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AP Calculus AB/BC. 3.2 Differentiability, p. 109 Day 1. To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at:. Cusp – slopes of the secant lines approach −∞ from one side and ∞ from the other side. Corner – one-sided derivatives differ.

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ap calculus ab bc

AP Calculus AB/BC

3.2 Differentiability, p. 109

Day 1

slide2

To be differentiable, a function must be continuous and smooth.

Derivatives will fail to exist at:

Cusp – slopes of the secant lines approach −∞ from one side and ∞ from the other side.

Corner – one-sided derivatives differ.

slide3

Vertical tangent – the slopes of the secant lines approach −∞ or ∞ from both sides.

Discontinuity – One of the one-sided derivatives is nonexistent.

slide5

Example 1: The function below fails to be differentiable at x = 0. Tell whether the problem is a corner, a cusp, a verticaltangent, or a discontinuity.

Since we want to differentiate at x = 0, we know at x = 0, y = 1 by the definition.

But, tan−10 = 0. So, therefore the problem is discontinuity because there are two values of y at x = 0.

slide6

Derivatives on a Calculator, p. 111

For small values of h, the difference quotient often gives a good approximation of f′(x).

The graphing calculator uses nDERIV to calculate the numerical derivative of f.

A better approximation is the symmetric difference quotient which the graphing calculator uses.

slide7

Example 2: Computing a Numerical Derivative

Compute the numerical derivative of f(x) = 4x – x2 at x = 0.

Use h = 0.001.

4(x−h)

(x−h)2

(x + h)2

4(x + h)

= 4

slide8

Derivatives on the TI-83/TI-84/TI-89:

You must be able to calculate derivatives with the calculator and without.

So, you need to do them by hand when called for.

Remember that half the test is no calculator.

slide9

ENTER

MATH

Find at x = 2.

Example 3:

8: nDeriv( x ^ 3, x, 2, 0.001)

returns

Note: nDeriv( x ^ 3, x, 2, 0.001) is the same as

nDeriv( x ^ 3, x, 2).

slide10

Warning:

The calculator may return an incorrect value if you evaluate a derivative at a point where the function is not differentiable.

Examples:

nDeriv( 1/x, x, 0)

returns

nDeriv( abs(x), x, 0)

returns

slide11

Graphing Derivatives

Graph:

−10 ≤x≤ 10

−10 ≤ y ≤ 10

What does the graph look like?

This looks like:

Use your calculator to evaluate:

p

ap calculus ab bc1

AP Calculus AB/BC

3.2 Differentiability, p. 109

Day 2

slide13

There are two theorems on page 113:

Theorem 1: Differentiability Implies Continuity

If f has a derivative at x = a, then f is continuous at x = a.

Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

slide14

Between a and b, must take on every value between and .

Theorem 2: Intermediate Value Theorem for Derivatives

If a and b are any two points in an interval on which f is differentiable, then f ′ takes on every value between f′(a) and f′(b).

Example 1

slide15

Example 2: Find all values of x for which the function is differentiable.

For this particular rational function, factor the denominator.

The zeros are at x = 5 and x = −1 which is where the function is undefined.

Since f(x) is a rational function, it is differentiable for all values of x in its domain. Therefore, f(x) is not differentiable at x = 5 or x = −1 since 5 and −1 are not in the domain of f(x).

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