Flashcards in Key knowledge for Core Deck (37):
What happens in differentiation?
You times the coefficient by the power and then subtract one from the power.
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)
UV' + U'V
What differentiation rule would you use to work out the differential of the following equation, y = e^(2x+1)
(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)
f(x) = U/V
f'(x) = (VU' - UV')/(V^2)
What are the ways in which you can solve a quadratic equation?
- Quadratic formula
- Completing the square
What is the shape of a negative quadratic equation
What is the shape of a positive quadratic equation
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.
(-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?
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?
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?
Work out the range of k, when f(x) = k has no solutions and f(x) = (2√5)cos(2θ - 26.7).
1) Refer to cos(x) graph, which has asymptotes x = ±1
2) cos(2θ - 26.7) has no effect on the asymptotes, but the 2√5 does
3) the new asymptotes would be x = ±2√5
4) So the range of values to which have no solutions are k > 2√5 and k < -2√5
When you are told to find the equation of the circle, and are presented with an equation such as
x^2 + y^2 + 4x - 2y - 11 = 0
What would you do?
1) Group the x's together (x^2 + 4x) and the y's together (y^2 - 2y)
2) Complete the square for both x and y
3) Write in the form (x±a)^2 + (y±b)^2 = r^2
What is the inverse function of e^x?
To work out the value of α in
Rcos(2ө - α) = 4cos(2ө) + 2 sin(2ө)
What would you do?
1) cos(2ө - α) = cos(2ө)cos(α) + sin(2ө)sin(α)
2) Equate coefficients, so cos(α) = 4 and sin(α) = 2
3) Form tan(α), by dividing sin(α) by cos(α).
Hence tan(α) = 1/2
4) Solve tan(α) = 1/2, α = (tan^(-1)(1/2) = 26.6 degrees
What would you do to find the cartesian equation of the following parametric equations?
x = 2t
y = t^2
1) Make t the subject of one of the equations, (the best one to use in this case would be x =2t)
2) t = x/2, Substitute t into the y equation and simplify.
y = (x/2)^2 => y = (x^2) / 4
To find the area under a curve given parametrically, what formula do you need to use?
x = 5t^2
y = t^3
The formula you need to remember is ∫y (dx/dt) dt
y = t^3
dx/dt = 10t
∫(t^3)(10t) = 10t^4
How do you find the gradient on the curve that is given parametrically?
Find the gradient at the point P where t = 2, on the curve given by x = t^3 + t and y = t^2 + 1.
1) Chain rule, so dy/dx = (dy/dt) X (dt/dx)
2) dy/dt = 2t
3) dx/dt = 3t^2 + 1 => dt/dx = 1/((3t^2) + 1)
4) dy/dx = (2t) X (1/((3t^2) + 1)) = 2t/((3t^2) + 1)
5) Substitute t = 2, and solve
6) dy/dx = 4/13
dt/dx = 1/(dx/dt)
What would be the answer to the differential of y^3?
What would be the answer to the differential of
y^2 + y
2y(dy/dx) + 1(dy/dx)
Differentiate the implicit equation
x^3 + x + y^3 + 3y = 6
3x^2 + 1 + 3y^2(dy/dx) + 3(dy/dx) = 0
In vectors how do you work out IaI?
a = i + 2j + 4k
IaI = √(1)^2 + (2)^2 + (4)^2
IaI = √1+4+16 = √21
If you the non-zero vectors a and b are perpendicular, what is a.b equal to?
a.b = 0