Key knowledge for Core Flashcards
What happens in differentiation?
You times the coefficient by the power and then subtract one from the power.
Eg.
f(x) = 3x^4
f’(x) = 12x^3
What happens in integration?
You add one to the power and then divide the coefficient by the new power.
f’(x) = 12x^3
f(x) = (12x^4) / 4 = 3x^4
When an equation has no real roots, how would you show this using the discriminant?
b^2 - 4ac < 0
When an equation has real roots, how would you show this using the discriminant?
b^2 - 4ac > 0
When an equation has repeated roots, how would you show this using the discriminant?
b^2 - 4ac = 0
If you had to work out the value of r is a geometric series, but you only had the third and fifth term?
third term = x
fifth term = y
(ar^4)/(ar^2) = y/x
simplifies to r^2 = y/x
r = √(y/x)
How would you cancel out e^x?
Times everything by ln.
How would you cancel out ln(x)?
Times everything by e.
What differentiation rule would you use to work out the differential of the following equation, y = x^(2)sin(2x)
Product rule.
UV’ + U’V
What differentiation rule would you use to work out the differential of the following equation, y = e^(2x+1)
Chain rule.
dy/du) x (du/dx
What differentiation rule would you use to work out the differential of the following equation, y = (2x+1)/(3x+7)
Quotient rule.
f(x) = U/V
f’(x) = (VU’ - UV’)/(V^2)
What are the ways in which you can solve a quadratic equation?
- Factorising
- Quadratic formula
- Completing the square
What is the shape of a negative quadratic equation
(-x^2)?
∩
What is the shape of a positive quadratic equation
(x^2)?
∪
In simultaneous equations if you have a linear and a quadratic equation, what would you do to solve it?
Substitute the linear equation into the quadratic equation.
When you multiply or divide an inequality by a negative number, what do you need to do?
Change the inequality sign to its opposite.
Eg.
(-y > 3) x (-1) = y < -3
If you have an equation of a line and need to find the equation of the normal to that line, what do you need to do to the gradient?
-(1/m)
Change the sign of the gradient and flip it.
If you had a question such as divide (x^3 - 1) by (x - 1), what would you need to include to make it possible?
0x^2 + 0x
………….______________
(x - 1) √x^3 + 0x^2 + 0x - 1
When using the factor theorem, if you are told f(a) is a factor, what should the equation equal when you substitute it in?
0
eg.
You are told that (x-2) is a factor of x^3 + x^2 - 4x - 4, so substitute,(x - 2 = 0 => x = 2), f(2) = (2)^3 + (2)^2 - 4(2) - 4 => 0, hence it is a factor.
If you are told to solve the following equation, what would you do?
3^(2x) + 3^(x+1) - 10 = 0
1) State y = 3^x
2) 3^(2x) = (3^x)(3^x) and 3^(x+1) = (3^x)(3^1)
3) Substitute y into the equation
4) y^2 + 3y - 10 = 0
5) Then solve
When using binomial expansion on an equation such as (3 + x)^4, what do you need to do?
Factor out 3 so you form (1 + x/3)^4, but you need to remember to put whatever you factor out to the power of the expansion, so it should be (3^4)(1 + x/3)^4
How do you show if a point is a maximum point?
d^(2)y/dx^(2) < 0
How do you show if a point is a minimum point?
d^(2)y/dx^(2) > 0
What is 1/sec(x) equivalent to?
cos(x)