Download Advanced Calculus Differentiation and Integration with Series

Advanced Calculus
Differentiation and Integration with Series
Examples:
1. If f (x) = 1+ 3(x + 2) +
5
7
(x + 2)2 + (x + 2)3 +....+ what is the value of f ¢¢(-2)?
2!
3!
2. Let f(x) be a function that is differentiable for all x. The derivative of this function is given by
9x 3 81x5 3x 7
+
+... If f(0) = 2, then f(x)=
the power series f ¢(x) = 3x 2
40
60
Homework:
n
4
1
1
1
1
2
3
n (x - 2)
find f ¢¢¢(2)
1. If f (x) = 1- (x - 2)+ (x - 2) - (x - 2) + ( x - 2) +...+ (-1)
3
9
27
81
3n
2. The Taylor Series of a function f(x) about x = 3 is given by
3(x - 3) 5(x - 3)2 7(x - 3)3
(2n+1)(x - 3)n
f (x) =1+
+
+
+...+
+... What is the value of f ¢¢¢(3)?
1!
2!
3!
n!
( x - 4) - ( x - 4) + ( x - 4) + 2 .
3. The third-degree Taylor polynomial for a function f about x = 4 is
3
What is the value of f ¢¢¢(4)?
512
2
64
4
4. Let f(x) be a function that is differentiable for all x. The derivative of this function is given by
8x 3 10x 4
+
+... If f(0) = 11, then f(x)=
the power series f ¢(x) = 4x + 2x 2 +
5
7
5. Let f(x) be a function that is differentiable for all x. The derivative of this function is given by
2(x +1) 3(x +1)2 4(x +1)3 5(x +1)4
+
+
+
+... If f(-1) = 4, then f(x)=
the power series f ¢(x) =
3
4
5
6