C3 Flashcards
What is domain of function
x values ↔ e.g -1
What is the range of a function
y values ↨ e.g f(x) > -5
What is the order for function fg(x)
First g then f = f [ g(x) ]
When do functions have an inverse
when f(x) is one-one
How do you find inverse equation
- Rearrange to make x the subject
2. Switch y and x
What is the range and domain for inverse functions
Inverse domain= normal range
Inverse range= normal domain
How do you solve modulus equations e.g | 9-x^2 |=5
- First solve equation removing modulus sign
2. Then switch the sign of the modulus parts and solve it again.
How do you make modulus graphs on calculator
- Graph section
- OPTN
- Num
- ABS
- Brackets around the equations.
Describe the inverse graph sin-1(x)
- domain -1< x <1
- range -pi/2 < sin^-1(x) < pi/2
- < should be the same or equal to sign
Describe the inverse graph tan^-1(x)
- domain- all real numbers
2. Range -pi/2 < tan-1(x)
Describe the inverse graph cos-1 (x)
- Domain -1< x <1
- Range 0< cos^-1(x)< pi
- < should be the same or equal to sign
What is 1/sin(x) and describe the graph
- cosec(x)
- The graph is separate curves that start at the points where the sin(x) meet 1 and -1.
- Asymptotes where sin (x) graph met x-axis
What is 1/cos(x) and describe the graph
- Sec (x)
- The graph is separate curves that start at the points where the cos(x) meet 1 and -1.
- Asymptotes where cos (x) graph met x-axis
What is 1/tan(x) and describe the graph
- Cot(x)
- reflections of the tan(x) graph
- Look at it as it is hard to describe
What is the relationship between e^x and ln(x)
They are inverse functions so cancel each other out
Describe the graphs of ln(x) and e^x
- e^x looks like a graph of any number ^x with an asymptote at y=1
- ln(x) is the reflection y=x with asymptote at x=1
Differentiate e^x
e^x
Differentiate e^kx
ke^kx
Differentiate ln(x)
1/x
What is the circle of differentiation starting with sin(x)
sin(x) → cos(x) → -sin(x) → -cos(x) → sin(x)
Differentiate tan(x) and tan(ax)
sec^2(x), asec^2 (ax)
Differentiate cot(x) and cot(ax)
-cosec^2(x), -acosec^2(ax)
Differentiate sec(x) and sec(ax)
sec(x)tan(x), asec(ax)tan(ax)
Differentiate cosec(x) and cosec(ax)
-cosec(x)cot(x), -acosec(ax)cot(ax)
Differentiate sin(ax) and cos(ax)
acos(ax), -asin(ax)
Differentiate ln(f(x))
f’(x)/ f(x)
What are the 3 rules for differentiation
- Product rule y=uv
- Quotient rule y=u/v
- Chain rule, used for composite functions e.g y=(2x+4)^3, y= sin(x+1)
Write formula of product rule
dy/dx= u* dv/dx + v*du/dx
Write formula of quotient rule
dy/dx= (Vdu/dx - Udv/dx)/V^2
Write formula of chain rule
dy/dx= dy/du * du/dx
How do you show equation x^3 + x^2 -5 =0 has root between x=1 and x=2
Do f(1) and f(2) and if the sign changes and functions it shows the root is between 1 and 2
volume of revolution formula about x axis
Volume= integration( pi*y^2) dx
Volume of revolution formula about y axis
Volume= integration( pi*x^2) dy
