Key knowledge for Core Flashcards by Connor Cullinane

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

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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

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When an equation has no real roots, how would you show this using the discriminant?

b^2 - 4ac < 0

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When an equation has real roots, how would you show this using the discriminant?

b^2 - 4ac > 0

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When an equation has repeated roots, how would you show this using the discriminant?

b^2 - 4ac = 0

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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)

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How would you cancel out e^x?

Times everything by ln.

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How would you cancel out ln(x)?

Times everything by e.

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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

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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

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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)

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What are the ways in which you can solve a quadratic equation?

Factorising
Quadratic formula
Completing the square

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What is the shape of a negative quadratic equation

(-x^2)?

∩

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What is the shape of a positive quadratic equation

(x^2)?

∪

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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.

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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)

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?

ln(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? Eg. x = 5t^2 y = t^3

The formula you need to remember is ∫y (dx/dt) dt Eg. y = t^3 dx/dt = 10t ∫(t^3)(10t) = 10t^4

How do you find the gradient on the curve that is given parametrically? Eg. 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?

3y^2(dy/dx)

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? eg. 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

If you the non-zero vectors a and b are parallel, what is a.b equal to?

a.b = IaI IbI