BASIC CALCULUS QUICK NOTES Page 2
6. DIFFERENTIATION
1. What is Calculus ? The derivative of a function tells how fast
Key Idea
Calculus is the branch of mathematics that it is changing at any point.
Differentioation is also called the slope of the curve.
studies change and accumulation. How fast
It has to main parts: things change 7. Basic Rules
Differentiation (rate of change) Integration
• Integration (total accumulation) Total Quick Trick
amount
() d (*) - nz (Power Rule)
• Bring the power
2. Real Life Applications d (k) 0 (k is constant) down.
4. Graph of a Function do
. Find velocitų and acceleration
Calculate area and volume Let f(x) 2x + 3 (u) (kx) k (k is Constant) • Reduce the
• Model population growth exponent by 1.
Analyze cost and revenue x-2 -1 1 2 3 (iw) (utv)
flx)-1 13 5 74
3. Functions
8. Examples
A function gives exactly one output 9
Fy2x*3
for cach input.
5
(i) y x' (iv) y = 9
Example: f(x) 2x + 3 dy dy
dx
If x = 5, f(5) 2(5) + 3 13 1 (@) y 4x + 3x2 (v) y 6x
Input - Function → Output dy = 16x + 6x dy
1 2 X
(ü) y = 5x- 2x
5. Limits (Basie Idea)
d - 15x - 2
A limit shows the value that a function
approaches as the input qets closer
to a certain number. Tip! 9. Slope from a Graph
We write lim f(x) L Caleulus is all
For y = x*, the slope changes as x hanges. 1.
about change At x = 0, slope = 0 (flat)
It means f(x) approaches L as x and accumulation
At x 1, slope 2 (positive)
approaches a.
-2 -1 O| 1 2*
Example : lim (3x + 2) 1
x3
At x =1, slope -2 (negative)
(because 3(3) + 2 11)
10. INTEGRATION
14. PRACTICE QUESTIONS
Integration is the reverse process of differentiation.
It helps us find total area, total amount, FORMULA SHEET
or original function.
O Differentiale: y x
Differentiate : 4z+ 7x: (") nx
11. Basic Rules
Quick Trick
3) Differentiate: y 10x-3x (k) - o
+ C (n-1) Increase the
n+1
power by 1, Find the slope of y x* at x 3. (kx) k
(i) [ k dx • kx +c (k is conskant) then divide by
6) Integrate
the new power.
+ C at Uhe end. 6 Inteqrate
• (utv)
(iw) ku dx k fu dz
O Integrate : (6x + 5) dx
12. Examples
8) A car's position is s 3x.
() Sdz c 4X+ 1
Find the velocity function v .
) Sketch the graph of y x*- 3.
10 Sketch the graph of y 2x + 4.
(iv) [9dx 9x + C
13. Area Under a Curve QuIcK MEMORY
The definite integral gives the area under the
curve between two points. Differentiation → Slope (rate of change)
Area = f(*) dx Integration Area (tokal amount)
• Derivative Bring power down
Example: Find area under y= 2x + 1 from x 1 to x4. Inerease power by 1, then divide
Integration
• Don't forget + C in integration !
- (4'+ 4) - (1 1)
= (16 + 4) - (1+ 1) = 20- 2 = 18 square units.
6. DIFFERENTIATION
1. What is Calculus ? The derivative of a function tells how fast
Key Idea
Calculus is the branch of mathematics that it is changing at any point.
Differentioation is also called the slope of the curve.
studies change and accumulation. How fast
It has to main parts: things change 7. Basic Rules
Differentiation (rate of change) Integration
• Integration (total accumulation) Total Quick Trick
amount
() d (*) - nz (Power Rule)
• Bring the power
2. Real Life Applications d (k) 0 (k is constant) down.
4. Graph of a Function do
. Find velocitų and acceleration
Calculate area and volume Let f(x) 2x + 3 (u) (kx) k (k is Constant) • Reduce the
• Model population growth exponent by 1.
Analyze cost and revenue x-2 -1 1 2 3 (iw) (utv)
flx)-1 13 5 74
3. Functions
8. Examples
A function gives exactly one output 9
Fy2x*3
for cach input.
5
(i) y x' (iv) y = 9
Example: f(x) 2x + 3 dy dy
dx
If x = 5, f(5) 2(5) + 3 13 1 (@) y 4x + 3x2 (v) y 6x
Input - Function → Output dy = 16x + 6x dy
1 2 X
(ü) y = 5x- 2x
5. Limits (Basie Idea)
d - 15x - 2
A limit shows the value that a function
approaches as the input qets closer
to a certain number. Tip! 9. Slope from a Graph
We write lim f(x) L Caleulus is all
For y = x*, the slope changes as x hanges. 1.
about change At x = 0, slope = 0 (flat)
It means f(x) approaches L as x and accumulation
At x 1, slope 2 (positive)
approaches a.
-2 -1 O| 1 2*
Example : lim (3x + 2) 1
x3
At x =1, slope -2 (negative)
(because 3(3) + 2 11)
10. INTEGRATION
14. PRACTICE QUESTIONS
Integration is the reverse process of differentiation.
It helps us find total area, total amount, FORMULA SHEET
or original function.
O Differentiale: y x
Differentiate : 4z+ 7x: (") nx
11. Basic Rules
Quick Trick
3) Differentiate: y 10x-3x (k) - o
+ C (n-1) Increase the
n+1
power by 1, Find the slope of y x* at x 3. (kx) k
(i) [ k dx • kx +c (k is conskant) then divide by
6) Integrate
the new power.
+ C at Uhe end. 6 Inteqrate
• (utv)
(iw) ku dx k fu dz
O Integrate : (6x + 5) dx
12. Examples
8) A car's position is s 3x.
() Sdz c 4X+ 1
Find the velocity function v .
) Sketch the graph of y x*- 3.
10 Sketch the graph of y 2x + 4.
(iv) [9dx 9x + C
13. Area Under a Curve QuIcK MEMORY
The definite integral gives the area under the
curve between two points. Differentiation → Slope (rate of change)
Area = f(*) dx Integration Area (tokal amount)
• Derivative Bring power down
Example: Find area under y= 2x + 1 from x 1 to x4. Inerease power by 1, then divide
Integration
• Don't forget + C in integration !
- (4'+ 4) - (1 1)
= (16 + 4) - (1+ 1) = 20- 2 = 18 square units.
