#Fundamental Theorem of Calculus

#Table of Contents

Introduction
First Fundamental Theorem of Calculus
Second Fundamental Theorem of Calculus
Worked Example
Practice Questions
Glossary
Summary and Key Takeaways

#Introduction

Key Concept

The Fundamental Theorem of Calculus is a crucial concept in calculus that links the concepts of differentiation and integration. It is divided into two parts:

The First Fundamental Theorem of Calculus
The Second Fundamental Theorem of Calculus

#First Fundamental Theorem of Calculus

#Definition

The First Fundamental Theorem of Calculus connects antiderivatives with the value of definite integrals. This theorem provides a straightforward method for evaluating definite integrals.

#Theorem Statement

If $f$ is a function that is continuous on the closed interval $[a, b]$ , and $F$ is an antiderivative of $f$ on $[a, b]$ , then:

$\int_{a}^{b} f(x) , dx = F(b) - F(a)$

Exam Tip

Be sure to review how to evaluate definite integrals using this theorem. Refer to the 'Evaluating Definite Integrals' study guide for detailed steps.

#Second Fundamental Theorem of Calculus

#Definition

The Second Fundamental Theorem of Calculus allows an antiderivative to be expressed as an accumulation function, where the variable $x$ appears as an integration limit.

#Theorem Statement

If $f$ is a function that is continuous on an interval containing $a$ , then for values of $x$ in that interval:

$\frac{d}{dx} \left( \int_{a}^{x} f(t) , dt \right) = f(x)$

Exam Tip

Exam questions may not explicitly reference this theorem but will expect you to recognize and apply its implications.

#### Example: Given the function

h(x) = \int\_{0}^{x} f(t) \, dt

, determine which of the following is true: (A)

h(4) < h'(4) < h''(4)

(B)

h(4) < h''(4) < h'(4)

(C)

h'(4) < h(4) < h''(4)

(D)

h''(4) < h(4) < h'(4)

(E)

h''(4) < h'(4) < h(4)

Solution:

$h(4) = \int_{0}^{4} f(t) , dt$ : If $f$ is positive over the interval $[0, 4]$ , then $h(4) > 0$ .
$h'(x) = \frac{d}{dx} \left( \int_{0}^{x} f(t) , dt \right) = f(x)$ by the Second Fundamental Theorem. Therefore, $h'(4) = f(4)$ . If $f(4) = 0$ , then $h'(4) = 0$ .
$h''(x) = \frac{d}{dx} \left( h'(x) \right) = f'(x)$ . If $f$ is decreasing at $x=4$ , then $f'(4) < 0$ , hence $h''(4) < 0$ .

Thus, $h''(4) < h'(4) < h(4)$ , so the correct option is (E).

#Practice Questions

Practice Question

#Question 1:

Evaluate the integral $\int_{1}^{3} (2x^2 - 1) , dx$ using the First Fundamental Theorem of Calculus.

Practice Question

#Question 2:

Given $g(x) = \int_{2}^{x} e^{t^2} , dt$ , find $g'(x)$ using the Second Fundamental Theorem of Calculus.

#Glossary

Antiderivative: A function $F(x)$ such that $F'(x) = f(x)$ .
Continuous Function: A function without any breaks, jumps, or holes in its domain.
Definite Integral: An integral evaluated over a specific interval $[a, b]$ .
Accumulation Function: A function that represents the accumulated area under a curve from a fixed point to a variable point.

#Summary and Key Takeaways

The First Fundamental Theorem of Calculus connects antiderivatives to definite integrals, providing a method to evaluate integrals.
The Second Fundamental Theorem of Calculus expresses the derivative of an accumulation function in terms of the original function.
Understanding these theorems is crucial for solving problems involving integration and differentiation.

#Exam Strategy

Familiarize yourself with the implications of both theorems.
Practice evaluating integrals and derivatives using these theorems.
Recognize problems that implicitly require the use of the Fundamental Theorem of Calculus.

Exam Tip

This content covers key objectives from the IB curriculum, specifically the links between differentiation and integration.