Maths 1 Week 9 Summary

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Important Concepts

Critical Points

Critical points of a function are points where the derivative is zero or undefined. These points are significant because they can indicate local maxima, local minima, or saddle points. Mathematically, if \( f'(x) = 0 \) or \( f'(x) \) does not exist at \( x = c \), then \( c \) is a critical point.

Local Minima and Local Maxima

A function \( f(x) \) has a local maximum at \( x = c \) if \( f(c) \) is greater than or equal to \( f(x) \) for all \( x \) in some interval around \( c \). Similarly, \( f(x) \) has a local minimum at \( x = c \) if \( f(c) \) is less than or equal to \( f(x) \) for all \( x \) in some interval around \( c \).

To find local maxima and minima, we use the first derivative test. If \( f'(x) \) changes sign from positive to negative at \( x = c \), then \( f(c) \) is a local maximum. If \( f'(x) \) changes sign from negative to positive at \( x = c \), then \( f(c) \) is a local minimum.

Increasing and Decreasing Functions

A function \( f(x) \) is increasing on an interval if \( f'(x) > 0 \) for all \( x \) in that interval. Conversely, \( f(x) \) is decreasing on an interval if \( f'(x) < 0 \) for all \( x \) in that interval.

Saddle Points

A saddle point is a critical point that is not a local maximum or minimum. At a saddle point, the function changes direction, but it does not reach a peak or a trough. For example, the function \( f(x, y) = x^2 - y^2 \) has a saddle point at \( (0, 0) \).

Second Derivative Test

The second derivative test helps determine whether a critical point is a local maximum, local minimum, or saddle point. If \( f''(x) > 0 \) at a critical point \( x = c \), then \( f(c) \) is a local minimum. If \( f''(x) < 0 \) at a critical point \( x = c \), then \( f(c) \) is a local maximum. If \( f''(x) = 0 \), the test is inconclusive.

Global Minima and Global Maxima

Global minima and maxima are the lowest and highest values of a function over its entire domain. A function \( f(x) \) has a global maximum at \( x = c \) if \( f(c) \geq f(x) \) for all \( x \) in the domain of \( f \). Similarly, \( f(x) \) has a global minimum at \( x = c \) if \( f(c) \leq f(x) \) for all \( x \) in the domain of \( f \).

Area of a Rectangle

The area \( A \) of a rectangle is given by the product of its length \( l \) and width \( w \):

$$ A = l \times w $$

Riemann Sum

A Riemann sum is a method for approximating the total area under a curve. It is calculated by dividing the region into small rectangles and summing their areas. Mathematically, a Riemann sum is given by:

$$ \sum_{i=1}^{n} f(x_i^*) \Delta x $$

where \( \Delta x \) is the width of each subinterval and \( x_i^* \) is a sample point in the \( i \)-th subinterval.

For example, to find the left Riemann sum for \( f(x) = x^2 \) on the interval \([0, 2]\) with \( n = 4 \) subintervals, we have:

$$ \Delta x = \frac{2-0}{4} = 0.5 $$

The left endpoints are \( x_0 = 0, x_1 = 0.5, x_2 = 1, x_3 = 1.5 \). The left Riemann sum is:

$$ \sum_{i=0}^{3} f(x_i) \Delta x = (0^2 \cdot 0.5) + (0.5^2 \cdot 0.5) + (1^2 \cdot 0.5) + (1.5^2 \cdot 0.5) = 0 + 0.125 + 0.5 + 1.125 = 1.75 $$

For the right Riemann sum, the right endpoints are \( x_1 = 0.5, x_2 = 1, x_3 = 1.5, x_4 = 2 \). The right Riemann sum is:

$$ \sum_{i=1}^{4} f(x_i) \Delta x = (0.5^2 \cdot 0.5) + (1^2 \cdot 0.5) + (1.5^2 \cdot 0.5) + (2^2 \cdot 0.5) = 0.125 + 0.5 + 1.125 + 2 = 3.75 $$

Riemann Integral

The Riemann integral is the limit of the Riemann sum as the width of the subintervals approaches zero. It represents the exact area under a curve. The Riemann integral of a function \( f(x) \) from \( a \) to \( b \) is denoted by:

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

Antiderivative

An antiderivative of a function \( f(x) \) is a function \( F(x) \) such that \( F'(x) = f(x) \). The process of finding an antiderivative is called antidifferentiation or integration.

Indefinite Integral

The indefinite integral of a function \( f(x) \) is the set of all antiderivatives of \( f(x) \). It is denoted by:

$$ \int f(x) \, dx $$

The indefinite integral includes a constant of integration \( C \) because the derivative of a constant is zero.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation and integration. It has two parts:

1. The first part states that if \( F(x) \) is an antiderivative of \( f(x) \), then:

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

2. The second part states that if \( f(x) \) is continuous on \([a, b]\), then the function \( F(x) \) defined by:

$$ F(x) = \int_{a}^{x} f(t) \, dt $$

is an antiderivative of \( f(x) \).

Differentiation

$$\frac{d}{dx} x^n = nx^{n-1}$$

$$\frac{d}{dx} e^x = e^x$$

$$\frac{d}{dx} \ln x = \frac{1}{x}$$

$$\frac{d}{dx} \sin x = \cos x$$

$$\frac{d}{dx} \cos x = -\sin x$$

$$\frac{d}{dx} \tan x = \sec^2 x$$

$$\frac{d}{dx} \cot x = -\csc^2 x$$

$$\frac{d}{dx} \sec x = \sec x \tan x$$

$$\frac{d}{dx} \csc x = -\csc x \cot x$$

$$\frac{d}{dx} a^x = a^x \ln a$$

Integration

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$

$$\int e^x \, dx = e^x + C$$

$$\int \frac{1}{x} \, dx = \ln|x| + C$$

$$\int \sin x \, dx = -\cos x + C$$

$$\int \cos x \, dx = \sin x + C$$

$$\int \sec^2 x \, dx = \tan x + C$$

$$\int \csc^2 x \, dx = -\cot x + C$$

$$\int \sec x \tan x \, dx = \sec x + C$$

$$\int \csc x \cot x \, dx = -\csc x + C$$

$$\int a^x \, dx = \frac{a^x}{\ln a} + C \quad (a > 0, a \neq 1)$$