Calculus is a branch of mathematics that deals with the study of rates of change and accumulation. It has two major branches: differential calculus (concerned with derivatives) and integral calculus (concerned with integrals). Both branches are connected by the Fundamental Theorem of Calculus. In this lesson, we will explore the concepts of derivatives, integrals, and limits, along with examples to illustrate each concept.
A derivative measures how a function changes as its input changes. It represents the rate of change of a function with respect to a variable. If f(x) is a function, its derivative is denoted by f'(x) or df/dx.
The derivative of f(x) at a point x = a is defined as:
f'(a) = limh→0 (f(a + h) - f(a)) / h
Let's find the derivative of the polynomial function f(x) = x3 - 5x2 + 7x - 3.
Using the power rule:
d/dx (xn) = nxn-1
Applying this to each term in the polynomial:
f'(x) = d/dx (x3) - d/dx (5x2) + d/dx (7x) - d/dx (3)
f'(x) = 3x2 - 10x + 7
Consider the function f(x) = ex, where e is the base of the natural logarithm.
The derivative of ex is:
f'(x) = ex
1. Velocity and Acceleration: If s(t) represents the position of an object as a function of time t, then the derivative s'(t) represents the velocity, and the second derivative s''(t) represents the acceleration.
2. Optimization: Derivatives are used to find local maxima and minima of functions, which are useful in optimization problems in economics, engineering, and other fields.
An integral is a mathematical object that represents the accumulation of quantities. The process of finding an integral is called integration. Integrals can be definite or indefinite.
An indefinite integral, or antiderivative, of a function f(x) is a function F(x) such that F'(x) = f(x). It is denoted by:
∫ f(x) dx = F(x) + C
where C is the constant of integration.
Find the indefinite integral of f(x) = 3x2 - 10x + 7.
Using the power rule for integration:
∫ xn dx = xn+1 / (n+1)
∫ (3x2 - 10x + 7) dx = ∫ 3x2 dx - ∫ 10x dx + ∫ 7 dx
= 3 * (x3 / 3) - 10 * (x2 / 2) + 7x + C
= x3 - 5x2 + 7x + C
A definite integral represents the accumulation of quantities over a specific interval. It is denoted by:
∫ab f(x) dx
Find the definite integral of f(x) = x2 from x = 1 to x = 3.
∫13 x2 dx = [ x3 / 3 ]13
= (33 / 3) - (13 / 3)
= (27 / 3) - (1 / 3)
= 9 - 1/3
= 26/3
1. Area Under a Curve: Definite integrals are used to find the area under a curve.
2. Accumulated Quantity: Integrals represent accumulated quantities such as distance traveled, total work done, and total mass.
A limit describes the behavior of a function as its input approaches a certain value. Limits are foundational to the concepts of derivatives and integrals.
The limit of f(x) as x approaches a is denoted by:
limx→a f(x) = L
if for every ε > 0, there exists a δ > 0 such that 0 < |x - a| < δ implies |f(x) - L| < ε.
Find the limit of f(x) = x2 - 4 as x approaches 2.
limx→2 (x2 - 4) = 22 - 4 = 4 - 4 = 0
Find the limit of f(x) = 1/x as x approaches infinity.
limx→∞ 1/x = 0
Continuity: A function is continuous at a point if the limit exists and equals the function's value at that point.
Derivatives and Integrals: Limits are used to define the concepts of derivatives and integrals.
The Fundamental Theorem of Calculus links the concepts of derivatives and integrals. It has two parts:
If F is an antiderivative of f on an interval [a, b], then:
∫ab f(x) dx = F(b) - F(a)
If f is a continuous real-valued function on [a, b], and F is defined by:
F(x) = ∫ax f(t) dt
for x in [a, b], then F is continuous on [a, b] and differentiable on (a, b), and F'(x) = f(x).
Let f(x) = 3x2. Find the definite integral from 0 to 2 using the Fundamental Theorem of Calculus.
First, find the antiderivative F(x):
F(x) = ∫ 3x2 dx = x3 + C
Now, apply the Fundamental Theorem:
∫02 3x2 dx = F(2) - F(0)
= (23) - (03)
= 8 - 0
= 8