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ERJAEE, G. (20)

Lecture 4

Department

MathematicsCourse Code

MATH 2BProfessor

ERJAEE, G.Lecture

4This

**preview**shows half of the first page. to view the full**2 pages of the document.** MATH 2B - Lecture 4 - The Fundamental Theorem of Calculus

Fundamental Theorem

● It is termed fundamental because it provides the link between the two branches

of calculus: differentiation and integration

○ Anti-differentiation: Find F(x) such that F’(x)=f(x)

○ (Definite) Integration: Find area under curve y=f(x)

● We need to first think of integrals as functions. We fix the lower limit of a definite

integral to be a constant a and let the upper limit be variable. Thus if f is a

function defined on an interval containing a and x, then is a function(x) (t)dtg =∫

x

a

f

of x

○ The function g returns the net area under the curve y=f(x) from a up to x.

○ Recall the conventions from the previous section for how to understand

values and net area: in particular, net area is negative if either

■x<a

■(t)f< 0

First part of the F.T.C.

● Suppose that f is continuous on [a,b]. Then is continuous on [a,b],(x) (t)dtF =∫

x

a

f

differentiable on (a,b), and its derivative is (x) (x)F′=f

● We often write . In essence, FTC part 1 says that if you integrate(t)dt (x)

d

dx ∫

x

a

f=f

then differentiate you return to what you started with

Example

1. The error function is differentiable for all real numbers x, and

2. os(t)dt os(x)

d

dx ∫

x

3

c2=c2

3. Switching limits: If the variable limit is at the bottom of the integral, we need to

switch the limits, thus introducing a minus sign, before applying the Theorem

dt dt

d

dx ∫

2

x

esin t= − d

dx ∫

x

2

esin t= − esin x

4. Combining with the chain rule: If the variable limit is more complicated, then we

need to use the chain rule

● If we let then(x) (t)dtF =∫

x

3

3− 1 5

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