Download 2.2 Derivative of Polynomial Functions A Power Rule Consider the

Calculus and Vectors – How to get an A+
2.2 Derivative of Polynomial Functions
A Power Rule
Consider the power function: y = x n , x, n ∈ R .
Then:
c) ( x 5 )'
n −1
n
Ex 1. For each case, differentiate.
a) (1)'
b) (x)'
( x )' = nx
d n
x = nx n −1
dx
⎛1⎞
d) ⎜ ⎟
⎝x⎠
'
⎛ 1 ⎞
e) ⎜ 7 ⎟
⎝x ⎠
Some useful specific cases:
(1)' = 0
f) ( x )'
( x)' = 1
g) (3 x )'
( x )' =
1
'
h) ( 4 x 3 )'
2 x
i) ( x π )'
B Constant Function Rule
Let consider a constant function: f ( x) = c, c ∈ R .
Then:
(c)' = 0
d
c=0
dx
Ex 2. For each case, differentiate.
a) (−2)'
C Constant Multiple Rule
Let consider g ( x) = cf ( x) . Then:
Ex 3. Differentiate each expression:
a) − 2x 3
b) ( π )'
[cf ( x)]' = cf ' ( x)
d
d
f ( x)
[cf ( x)] = c
dx
dx
(cf )' = cf '
b)
c)
−3
x2
2x
Ex 4. Differentiate.
a) f ( x) = 2 − 3 x + 4 x 2 − 3 x 7
D Sum and Difference Rules
[ f ( x) ± g ( x)] = f ' ( x) ± g ' ( x)
d
d
d
[ f ( x) ± g ( x)] =
f ( x) ±
g ( x)
dx
dx
dx
( f ± g )' = f '± g '
2.2 Derivative of Polynomial Functions
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Calculus and Vectors – How to get an A+
b) g ( x) =
1
2
4
−
+
x x2 x5
c) h( x) =
x −3 x
x
Ex 5. Find the equation of the tangent line at the point
E Tangent Line
To find the equation of the tangent line at the point
1
P (1,0) to the graph of y = f ( x) = x 2 − .
P(a. f (a)) :
x
1. Find derivative function f ' ( x) .
2. Find the slope of the tangent line using:
m = f ' (a)
3. Use the slope-point formula to get the equation
of the tangent line:
y − f ( a ) = m( x − a )
Ex 6. Find the equation of the tangent line of the
slope m = 2 to the graph of y = f ( x) = 2 x . Graph
the function and the tangent line.
Ex 7. Find the points on the graph of y = f ( x) = 2 x 3 − 3x 2 + 1
where the tangent line is horizontal.
2.2 Derivative of Polynomial Functions
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Ex 8. Find the equation of the normal line to the curve
F Normal Line
2
If mT is the slope of the tangent line, then slope of
y = f ( x) = x + at P (1,3) .
x
the normal line m N is given by:
1
mN = −
mT
Ex 9. Analyze the differentiability of each function.
b) y = f ( x) = x 2 / 3
a) y = f ( x) = 3 x
c) y = f ( x) =| x − 3 |
2.2 Derivative of Polynomial Functions
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H Differentiability for piece-wise defined
functions
Let consider the piece-wise defined function:
⎧ f1 ( x), x < a
⎪
f ( x) = ⎨c, x = a
⎪ f ( x), x > a
⎩ 2
Ex 10. Analyze the differentiability of each function at x = 1 .
⎧⎪ x 2 , x ≤ 1
a) f ( x) = ⎨
⎪⎩2 x, x > 1
The function f is differentiable at x = a if:
(a) the function is continuous at x = a
(b) f1 ' (a) = f 2 ' (a) (the slope of the tangent line for
the left branch is equal to the slope of the
tangent line for the right branch).
⎧⎪ x 2 , x ≤ 1
b) a) f ( x) = ⎨
⎪⎩ x, x > 1
⎧⎪ x 2 , x ≤ 1
c) a) f ( x) = ⎨
⎪⎩2 x − 1, x > 1
Reading: Nelson Textbook, Pages 76-81
Homework: Nelson Textbook: Page 82 #2cdf, 3ce, 4aef, 5b, 6b, 7a, 8b, 9b, 11, 14, 17, 20, 25iii, 28
2.2 Derivative of Polynomial Functions
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