differential calculus 1.ppt, a procedure for finding the exact derivative directly from the for- mula of the function, without having to use graphical methods

a procedure for finding the exact derivative directly from the for- mula of the function, without having to use graphical methods

1. AP Calculus AB/BC
3.2 Differentiability, p. 109
Day 1

2. To be differentiable, a function must be continuousand
smooth.
Derivatives will fail to exist at:
Corner –one-sided
derivatives differ.
Cusp –slopes of the secant
lines approach −∞from one
side and ∞from the other side.f x x 
2
3
f x x 

3. Vertical tangent –the slopes
of the secant lines approach
−∞or ∞from both sides.
Discontinuity –One of
the one-sided derivatives
is nonexistent.
3
f x x 
1, 0
1, 0
x
fx
x


 

4. Most of the functions we study in calculus will be differentiable.

5. Example 1:The function below fails to be differentiable at x= 0.
Tell whether the problem is a corner, a cusp, a vertical
tangent, or a discontinuity.1
tan , 0
1, 0

 


xx
y
x
Since we want to differentiate at x= 0, we know at x= 0, y= 1
by the definition. 
But, tan
−1
0 = 0. So, therefore the problem is discontinuity
because there are two values of yat x= 0.

6. Derivatives on a Calculator, p. 111
For small values of h, the
difference quotient often gives a
good approximation of f′(x). f a h f a
h
The graphing calculator uses nDERIVto calculate the
numerical derivative of f.
  
nDERIV
2
  

f a h f a h
fa
h
A better approximation
is the symmetric
difference quotient
which the graphing
calculator uses.

7. Example 2: Computing a Numerical Derivative
Compute the numerical derivative of f(x) = 4x–x
2
at x= 0.
Use h= 0.001.
= 4     
 
22
4 0+0.001 0 0.001 4 0 0.001 0 0.001
nDERIV( ( ), 0) =
2 0.001
         
   
fx 
  
nDERIV
2
  

f a h f a h
fa
h 
4(x+ h) (x+ h)
2 4(x−h) (x−h)
2

8. Derivatives on the TI-83/TI-84/TI-89:
You mustbe 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.

9. 3
yx Example 3: Find at x = 2.dy
dx
8: nDeriv( x ^ 3, x, 2, 0.001)ENTER
returns12.000001
MATH
Note: nDeriv( x ^ 3, x, 2, 0.001)is the same as
nDeriv( x ^ 3, x, 2).

10. Warning:
The calculator may return an incorrect value if you
evaluate a derivative at a point where the function is not
differentiable.
Examples:
returns1000000
returns0
nDeriv( 1/x, x, 0)
nDeriv( abs(x), x, 0)

11. Graphing Derivatives
Graph:
What does the graph look like?
This looks like: yx
Use your calculator to evaluate:2
−10 ≤x≤ 10
−10 ≤ y≤ 102

xx
y 1nDeriv(Xabs(X)/2,X,X)y nDeriv(Xabs(X)/2,X,2)
p

12. AP Calculus AB/BC
3.2 Differentiability, p. 109
Day 2

13. There are two theorems on page 113:
Theorem 1: Differentiability Implies Continuity
If fhas a derivative at x = a, then fis continuous at x= a.
Since a function must be continuous to have a derivative,
if it has a derivative then it is continuous.

14. 
1
2
fa 3fb Between aand b, must take
on every value between and .f 1
2 3
Theorem 2: Intermediate Value Theorem for Derivatives
If aand bare any two points in an interval on which fis
differentiable, then f ′takes on every value between f′(a)
and f′(b).
Example 1

15. Example 2:Find all values of xfor which the function is
differentiable.
3
2
8
45



x
fx
xx
For this particular rational function, factor the denominator.
2
4 5 5 1    x x x x
The zeros are at x= 5and x= −1 which is where the function is
undefined.
Since f(x)is a rational function, it is differentiable for all
values of xin its domain. Therefore, f(x)is not differentiable
at x= 5or x= −1since 5and −1are not in the domain of f(x).
p