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 $f^{\prime }(x)=0$ or $f^{\prime }(x)$ 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 $f^{\prime }(x)$ changes sign from positive to negative at $x=c$, then $f(c)$ is a local maximum. If $f^{\prime }(x)$ 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 $f^{\prime }(x)>0$ for all $x$ in that interval. Conversely, $f(x)$ is decreasing on an interval if $f^{\prime }(x)<0$ 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 $f(c)\ge f(x)$ for all $x$ in the domain of $f$. Similarly, $f(x)$ has a global minimum at $x=c$ if $f(c)\le f(x)$ 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:

$$\sum _{i=1}^{n}f(x_i^{\ast })\mathrm{\Delta }x$$

where $\mathrm{\Delta }x$ is the width of each subinterval and $x_i^{\ast }$ 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:

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

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

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

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:

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

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 $F^{\prime }(x)=f(x)$. 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

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

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

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

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

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

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

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

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

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

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

Integration

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

$$\int e^{x}dx=e^x+C$$

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

$$\int \sin xdx=-\cos x+C$$

$$\int \cos xdx=\sin x+C$$

$$\int {\sec }^{2}xdx=\tan x+C$$

$$\int {\csc }^{2}xdx=-\cot x+C$$

$$\int \sec x\tan xdx=\sec x+C$$

$$\int \csc x\cot xdx=-\csc x+C$$

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