Calculus

Calculus

Introduction to Calculus: What is Derivative and Integral

This is a topic ties everything about (functions and graphs) together.

We're mainly concerned with two parts:

1. Derivative (Differentiation)
2. Integrals (Integration)

A derivative is equal to the rate of change. Most of the time we will use $dy \over dx$ to present how one variable changes with another. The derivative is the gradient of a tangent line.

But from the beginning, we are going to talk about where do you want the derivative, the other way to say is I will give you a point and then tell me what's the derivative or gradient for that point.

From the beginning here are the concepts you need for calculus.

1. "what is the derivative at x=n" you can understand as "what is the gradient when x=n".
2. "What is the integral at x=a to x=b" you can understand as "what is the area between the function and the x-axis from x=a to x=b".

Conclusion (from the beginning )

derivative = gradient of a tangent = rate of change.

Integral = area under the function.

Differentiation

Limits

$\lim _{x\to a}f(x)=L,$

Properties of Limits

1. Sum Rule: The limit of the sum of two functions is the sum of their limits : $\lim _{x\to c}(f(x)+g(x))=\lim _{x\to c}(f(x))+\lim _{x\to c}(g(x))$
2. Difference Rule: The limit of the difference of two functions is the difference of their limits : $\lim _{x\to c}(f(x)-g(x))=\lim _{x\to c}f(x)-\lim _{x\to c}g(x)$
3. Product Rule: The limit of a product of two functions is the product of their limits : $\lim _{x\to c}(f(x)*g(x))=\lim _{x\to c}f(x)*\lim _{x\to c}g(x)$
4. Constant Multiple Rule: The limit of a constant times a function is the constant times the limit of the function :

$\lim _{x\to c}(k*g(x))=k*\lim _{x\to c}g(x)$

1. Quotient Rule: The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero. $\lim _{x\to c}{(f(x) \over g(x))}={\lim _{x\to c}f(x) \over \lim _{x\to c}g(x)}$

Power Rule

$(cx^{n})'=c*n*x^{n-1}$

Derivative of Polynomial Functions

=Newton Derivative of Polynomial Functions=

1. The sum rule $(f+g)'=f'+g'$
2. The Difference Rule $(f-g)'=f'-g'$
3. The Product Rule $(f*g)'=f*g'+g*f'$
4. The Quotient Rule $({f \over g})'={(gf'-fg') \over g^{2}}$

=Leibniz Derivative of Polynomial Functions=

1. The sum rule ${d(f+g) \over dx}={df \over dx}+{dg \over dx}$
2. The Difference Rule ${d(f-g) \over dx}={df \over dx}-{dg \over dx}$
3. The Product Rule ${d(fg) \over dx}'=f{dg \over dx}+g{df \over dx}$
4. The Quotient Rule ${d({f \over g}) \over dx}={g{df \over dx}-f{dg \over dx} \over g^{2}}$

Derivative of Trigonometric Functions

1. $f(sinx)'=cos(x)$
2. $f(cosx)'=-sin(x)$
3. $f(tanx)'=sec^{2}(x)$
4. $f(cotx)'=-csc^{2}(x)$
5. $f(cscx)'=-csc(x)cot(x)$
6. $f(secx)'=sec(x)tan(x)$

Chain Rule

1. Newton Chain rule $[f(g(x))]'=f'(g(x))*g'(x)$
2. Leibniz Chain rule ${\frac {df(g(x))}{dx}}={{df(g)} \over dg}{dg \over dx}$

Derivatives of Logarithmic and Exponential Functions

1. Natural log of $x$ : $(ln(x))'={1 \over x}*(x)'$
2. log of $x$ : $(log_{a}x^{z})'=({lnx^{z} \over lna})'={1 \over ln_{a}}(lnx^{z})'={zx^{z-1} \over ln{a*x^{z}}}$

The Exponential Function

1. Exponential $e=1+1+{1 \over 2!}+{1 \over 3!}+{1 \over 4!}+{1 \over 5!}+{1 \over 6!}+{1 \over 7!}....$
2. Exponential Function $e^{x}=1+x+{x^{2} \over 2!}+{x^{3} \over 3!}+{x^{4} \over 4!}+{x^{5} \over 5!}+{x^{6} \over 6!}+{x^{7} \over 7!}...$
3. Exponential Function $(e^{x})'=e^{x}$

so when every you meet Exponential just reminds yourself it will not be changed by the derivatives.

Higher Derivatives

1. Second derivatives $f''(x)=(f'(x))'$

Newton Higher Derivatives

1. function$f(x)$
2. first derivative $f'(x)$
3. Second derivatives $f''(x)=(f'(x))'$
4. Third derivatives $f'''(x)=(f''(x))'$
5. 4th derivatives $f^{(4)}(x)=(f'''(x))'$

Leibniz Higher Derivatives

1. function $y=f(x)$
2. first derivative ${dy} \over {dx}$
3. second derivatives ${d^{2}}y \over {dx^{2}}$
4. third derivatives ${d^{3}}y \over {dx^{3}}$
5. 4th derivatives ${d^{4}}y \over {dx^{4}}$

Why do we need Higher Derivatives

From the zero derivative to sixth derivative there is a meaning on the graph

1. No derivative function $f(x)$ = Position
2. first derivative $f'(x)$ = Velocity
3. second derivative $f''(x)$ = Acceleration
4. third derivative $f'''(x)$ = Jerk
5. fourth derivative $f^{(4)}(x)$ = Snap
6. fifth derivative $f^{(5)}(x)$ = crackle/flounce
7. sixth derivative $f^{(6)}(x)$ = Pop

How to finding Local Maxima and Minima by Differentiation

Our example will be : $f(x)=x^{3}-12x+1$ And the graph will be the one at the side.

Then we will need to use the derivatives to find the two-point that has 0 slopes we will also call it Local maxima and Local minima

$f'(x)=3x^{2}-12$

$f'(x)=3x^{2}-12$

$3x^{2}=12$

$x^{2}=4$

Then we get x=2 and x=-2 so then we put them into the function.

$f(x)=2^{3}-(12*2)+1=-15$

$f(x)=-2^{3}-(12*-2)+1=17$

so the first point will be local maxima it will be at (-2,17) and then local minima (2,-15).

Graphing Functions and Their Derivatives

1. When a first derivative is positive, the original function is increasing.
2. When a first derivative is negative, the original function is decreasing.
   1. The derivative does not tell us if the function is positive or negative, only if increasing or decreasing.
3. When a second derivative is positive, the original function is concave up.
4. When a second derivative is negative, the original function is concave down.
5. When a first derivative hits zero from below the axis, the original function is at a(n) local minimum.
6. When a second derivative is zero, the original function is at a(n) inflection point.

Limits and L'Hospital's Rule

1. L'Hospital's Rule 1 ${\textstyle \lim _{x\to 0}{f(x) \over g(x)}=lim_{x\to 0}{f'(x) \over g'(x)}}$
2. L'Hospital's Rule 2 ${\textstyle \lim _{x\to \infty }{f(x) \over g(x)}=lim_{x\to \infty }{f'(x) \over g'(x)}}$

Integration

Mathematicians have been playing with the concept of integration for ages, but it was until Newton’s time that it was realized that integration and differentiation are inverse operations.

Integral

1. Indefinite Integral $\int f(x)\,dx=F(x)+c,$
2. sum rule of Indefinite Integral $\int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx$
3. The Difference Rule $\int [f(x)-g(x)]\,dx=\int f(x)\,dx-\int g(x)\,dx$
4. Indefinite Integral $\int x^{n}\,dx={x^{n+1} \over n+1}+c$
5. Natural log rule $\int {n \over x}\,dx={ln|x^{n}|}$
6. $\int a^{x}dx={a^{x} \over ln(a)}$
7. constant(constant can be pull out in the Indefinite Integral) $\int c*f(x)dx=c\int f(x)dx$

Properties of Indefinite Integral

sum rule of Indefinite Integral

1. sum rule of Indefinite Integral $\int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx$

1. The Difference Rule $\int [f(x)-g(x)]\,dx=\int f(x)\,dx-\int g(x)\,dx$

Integrals With Trigonometric Functions

1. $\int cosx\,dx=sinx+c$
2. $\int sinx\,dx=-cosx+c$
3. $\int {sec}^{2}x\,dx=tanx+c$
4. $\int {csc}^{2}x\,dx=-cotx+c$
5. $\int secx\,tanx\,dx=secx+c$
6. $\int cscx\,cotx\,dx=-cscx+c$

Integration By Parts

1. Performing Integration By Parts $\int f(x)g'(x)\,dx=f(x)g(x)-\int f'(x)g(x)\,dx$

1. Performing Integration By Parts $\int u\,dv=uv-\int v\,du$

All equations that you may need to know for integration

1. $\int x^{n}\,dx={x^{n+1} \over n+1}$
2. $\int 1 \over x\,dx={ln|x|}$
3. $\int a^{x}\,dx={a^{x} \over lna}$
4. $\int e^{x}\,dx=e^{x}$
5. $\int sinx\,dx=-cosx$
6. $\int cosx\,dx=sinx$
7. $\int secxtanx\,dx=secx$
8. $\int sec^{2}x\,dx=tanx$
9. $\int csc^{2}x\,dx=-cotx$
10. $\int \csc x\cot x\,dx=-cscx$
11. $\int secx\,dx=ln|secx+tanx|$
12. $\int cscx\,dx=ln|cscx-cotx|$
13. $\int tanx\,dx=ln|secx|$
14. $\int cotx\,dx=ln|sinx|$

Definite integrals area under a curve and above the curve

1. Definite Integral under and above the curve $\int _{a}^{b}f(x)\,dx,\int _{b}^{c}|f(x)|\,dx=F(b)-F(a)+|F(b)-F(a)|,$

Area between curves

$\int _{a}^{b}(upperfunction-lowerfunction)\,dx,$

if f(x) = upper function and g(x) = lower function

$\int _{a}^{b}(f(x)-g(x))\,dx,$

Kinematics

Kinematics is a type of physics

Kinematics is a physics topic taking about how describes the motion of points, objects without considering the forces that cause them to move. In Kinematics we will use calculus a lot of times, by using calculus can be a kind of transformation of the graph.

So it will be required Higher Derivatives and Integration.

derivative and integration

1. $s(t)=$ displacement (m)
2. first derivative $s'(t)=a(t)=$ velocity (m/s)
3. second derivative $s''(t)=v(t)=$ acceleration
4. third derivative $s'''=s(t)=$ time (second or s )

Same integration

1. no integration $t=$ time (second or s )
2. first integration of t $a(t)=$ acceleration
3. second integration of t $v(t)=$ velocity (m/s
4. third integration of t $s(t)=$ displacement (m)

Using Integration to calculate volume

This time we will start to learn how to use integration to calculate the volume.

Now we will start with a function:

$f(x)={\sqrt {x}}$

We will start to calculate the volume of the function when the function rotates 360 degrees on the x-axis.

It will never leave the x-axis, and then it will form a 3-dimensional volume. We will call this solid of revolution because we obtain it by revolving a region about a line.

$f(x)={\sqrt {x}}$ $\int _{0}^{1}A(x)dx$

so the radius of a circle will be r and $r={\sqrt {x}}$

and we can think of it as the volume is made of many disks and it will form this volume.

So the area of the disk will be like this $A=\pi r^{2}$

$r={\sqrt {x}}$

$r^{2}=x$

so

$A=\pi {({\sqrt {x}})}^{2}$

$A(x)=\pi x$

so what we will get this

$\int _{0}^{1}\pi xdx$

so $\pi$ is a constant so we can pull it out

$\pi \int _{0}^{1}xdx$

Reference

1. IB Maths AA SL/HL calculus
2. Calculus by Professor Dave Explains