Functions Defined Using a Definite Integral and The Second Fundamental Theorem of Calculus
Functions Defined Using a Definite Integral and The Second Fundamental Theorem of Calculus
Lesson objectives and vocabulary followed by...
0:38 Examples that Help Us to Understand How a Function May Be Defined Using a Definite Integral
4:51 A Function Defined Using a Definite Integral
6:42 Using the First Fundamental Theorem of Calculus to Find the Derivative of a Function Defined Using a Definite Integral
9:40 Second Example of Using the First Fundamental Theorem of Calculus to Find the Derivative of a Function Defined Using a Definite Integral
11:19 Third Example of Using the First Fundamental Theorem of Calculus to Find the Derivative of a Function Defined Using a Definite Integral
13:11 The Second Fundamental Theorem of Calculus
14:49 A Proof of the Second Fundamental Theorem of Calculus
EXAMPLES of Using the Second Fundamental Theorem of Calculus to Find Derivatives of Functions Defined Using a Definite Integral
16:08 Ex. 1) g(x) = The Definite Integral of cos t WRT t from t=0 to t=x
16:46 Ex. 2) g(x) = The Definite Integral of cos t WRT t from t=0 to t=x^2
17:10 Seeing g as a Composition of Two Functions
19:08 Using the Chain Rule to Find the Derivative of g
20:41 Ex. 3) g(x) = The Definite Integral of cos t WRT t from t=0 to t=x^3
20:52 Seeing g as a Composition of Two Functions
21:17 Using the Chain Rule to Find the Derivative of g
22:08 Ex. 4) g(x) = The Definite Integral of 2t^2 + 5 WRT t from t=0 to t=x
22:36 Ex. 5) g(x) = The Definite Integral of 2t^2 + 5 WRT t from t=0 to t=sin x
22:51 Seeing g as a Composition of Two Functions
23:55 Using the Chain Rule to Find the Derivative of g
24:57 Ex. 6) g(x) = The Definite Integral of the Square Root of (t^2 + 5) WRT t from t=cos x to t=5
25:36 Rewriting g so that the Lower Limit of Integration is a Constant
26:01 The Lower Limit of Integration CAN BE ANY CONSTANT (Not Just Zero)
26:14 Using the Chain Rule to Find the Derivative of g
27:42 Summary of Lesson
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