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Pure maths
| Question | Answer |
|---|---|
| n^a x n^b = | n^(a+b) |
| n^a / n^b = | n^(a-b) |
| (n^a)^b = | n^ab |
| a^0 = | 1 |
| a^(1/m) = | m√a |
| a^(n/m) = | (m√a)^n |
| a^-m = | 1/a^m |
| √a x √b = | √ab |
| √a / √b = | √(a/b) |
| What is the quadratic formula | (-b±√(b^2-4ac))/2a |
| Define domain | Set of possible inputs |
| Define range (in terms of functions) | Set of possible outputs |
| If a solution has no real roots then (discriminant) | b^2-4ac <0 |
| If a solution has equal roots then (discriminant) | b^2-4ac =0 |
| If a solution has two real roots then (discriminant) | b^2-4ac >0 |
| What does ∅ mean | The empty set |
| What does ∪ and ∩ mean | ∪ = or ∩ = and |
| What does ∈ mean | Is a member of |
| For transforming graphs, what does a change inside f(x) mean | Affects x in the opposite way to what we expect |
| For transforming graphs, what does a change outside f(x) mean | Affects y as expected |
| How does f(x-a) transform a graph | Translation (a,0) |
| How does f(x)+a transform a graph | Translation (0,a) |
| How does f(ax) transform a graph | Stretches horizontally Sf 1/a |
| How does af(x) transform a graph | Stretches vertically Sf a |
| When showing regions is < a solid or dashed line | Dashed |
| When showing regions is ≤ a solid or dashed line | Solid |
| What is the equation for gradient of a line | m=(y2-y1)/(x2-x1) |
| What is the equation to find equation of a line when you know the gradient | y-y1=m(x-x1) |
| What is the equation used to find distance between two points | √((x2-x1)+(y2-y1) |
| What is the equation for a circle around (a,b) | (x-a)^2 + (y-b)^2 =r^2 |
| What is it called when a triangle is inside a circle | Triangle inscribes the circle Circle circumscribes the triangle |
| If the circumscribing shape is a triangle what is the circle called | The circumcircle |
| What is the centre of the circumcircle called | the circumcentre |
| What is the factor theorem | If f(x) is a polynomial then if f(p)=0 then (x-p) is a factor of f(x) |
| How do we show an even number in proof | 2n |
| How do we show an odd number in proof | 2n+1 |
| How do we prove that something is positive | Complete the square |
| What does ≡ mean | Equivalent to |
| Define proof by exhaustion | Breaking down the statement into all possible smaller cases |
| What does n! mean | n x (n-1) x (n-2) x...x 2 x 1 |
| What does nCr mean | n!/r!(n-r)! |
| What is the binomial expansion | nCr a^(n-r) b^r |
| What is the cosine rule | a^2=b^2+c^2-2bc(cosA) |
| What is the sine rule | a/sinA=b/sinB=c/sinC |
| Sinθ= (Ambiguous case) | Sin(180-θ) |
| Cosθ= (Ambiguous case) | Cos(360-θ) |
| What is the area of a non right angled triangle | area=(1/2)ab(sinC) |
| Sin45= | 1/√2 |
| Cos45= | 1/√2 |
| Tan45= | 1 |
| Sin30= | 1/2 |
| Cos30= | √3/2 |
| Tan30= | 1/√3 |
| Sin60= | √3/2 |
| Cos60= | 1/2 |
| Tan60= | √3 |
| Sinθ/cosθ | tanθ |
| Sin^2(θ)+cos^2(θ)= | 1 |
| How often does sin and cos repeat | Every 360 degrees |
| How often does tan repeat | Every 180 degrees |
| What is a vector | A quantity with magnitude and direction |
| If vector PQ and vector RS then | The line segments PQ and RS are equal in length |
| AB+BC= (vectors) | AC |
| λ(pi+qj)= | λpi+λqj |
| (pi+qj)+(ri+sj)= | (p+r)i+(q+s)j |
| How do you calculate the magnitude of a vector | Pythagoras |
| What is the magnitude of a unit vector | 1 |
| How do you calculate a unit vector | Divide the vector by the magnitude |
| If the point P divides the line segment AB in the ratio λ:μ then OP= | OA+(λ/λ+μ)AB |
| If a and b are two non parallel vectors and pa+qb=ra+sb then | p=r and q=s |
| What is the equation for differentiating from first principals | f'(x)=lim(h->0)(f(x+h)-f(x))/h |
| If y=ax^n then dy/dx= | nax^(n-1) |
| differentiate 3 | 0 |
| What should you do if your expression isn't a sum of x^n terms | -Turn roots to powers -Split up fraction -Expand brackets -Beware of numbers in denominators |
| What is the equation of a tangent with coordinates (a,f(a)) | y-f(a)=f'(a)(x-a) |
| What is a normal | A line perpendicular to a tangent |
| For an increasing function m?0 | m≥0 |
| For a decreasing function m?0 | m≤0 |
| How do you prove that a function is always decreasing | Complete the square |
| If f''(a)>0 then it is a | Local min |
| If f''(a)<0 then it is a | Local max |
| If dy/dx =kx^n then y= | (k/n+1)x^(n+1)+c |
| How do we show that we are integrating with respect to x | ∫f(x)dx |
| How do we find c (Integration) | -Integrate -Substitute known point -Solve to find c |
| How do we do definite integration | -Write statement with limits -Integrate and write in square brackets -Evaluate the definite integral |
| What is the equation for area under a curve and the x-axis and the lines x=a and x=b | --------b Area=∫f(x)dx --------a |
| What should you do if a curve that you are finding the area of crosses the x-axis | Split into 2 separate integrations |
| What is the y intercept of y=2^x | 1 |
| y=e^x dy/dx= | e^x |
| What is the value of e | 2.71828 |
| What is the inverse of e | ln |
| loga(X)+loga(Y)= | loga(XY) |
| loga(X)-loga(Y)= | loga(X/Y) |
| loga(X)^k= | kloga(X) |
| loga(1/X)= | -loga(X) |
| loga(a)= | 1 |
| loga(1)= | 0 |
| How can you solve equations using logarithms | Taking logs of both sides |
| If f(x)=g(x) then logaf(x)= | logag(x) |
| What is ln | The inverse of e |
| How do we fit y=ax^n to y=mx+c | y=ax^n logy=logax^x logy=loga + logx^n logy=loga + nlogx |
| How do we fit y=ab^x to y=mx+c | y=ab^x logy=logab^x logy=loga + logb^x logy=loga + xlogb |
| How do we prove by contradiction | -Assume original statement is false -Reach a contradiction -Conclude original statement is true |
| How do we solve fg(x) | Substitute g(x) into f(x) |
| What is the equation for the nth term of an arithmetic sequence | Un=a+(n-1)d |
| What are the equations for sum of an arithmetic series | Sn=(n/2)(2a+(n-1)d) Sn=(n/2)(a+l) |
| What is the equation for the nth term of a geometric sequence | Un=ar^(n-1) |
| What is the equation for the sum of a geometric series | Sn=(a(1-(r^n)))/(1-r) |
| What is the equation for sum to infinity | S∞=a/(1-r) |
| When is a sequence converging | |r|<1 |
| A sequence is increasing if... | U(n+1)>Un for all n∈ℕ |
| A sequence is decreasing if... | U(n+1)<Un for all n∈ℕ |
| A sequence is periodic if... | The terms repeat in a cycle, U(n+k)=Un, k is the order |
| What is the binomial expansion of (1+x)^n where n∉ℕ | 1+nx+(n(n-1)/2!)(x^2)+(n(n-1)(n-2)/3!)(x^3)... |
| 2πrad= | 360deg |
| What is the equation for arc length | l=rθ |
| What is the equation for area of a sector | A=0.5(r^2)θ |
| What is the equation for area of a segment | A=0.5(r^2)(θ-sinθ) |
| When θ is small and measured in radians sinθ≈ | θ |
| When θ is small and measured in radians tanθ≈ | θ |
| When θ is small and measured in radians cosθ≈ | 1-(θ^2)/2 |
| secθ= | 1/cosθ |
| cosecθ= | 1/sinθ |
| cotθ= | 1/tanθ=cosθ/sinθ |
| 1+tan^2θ= | sec^2θ |
| 1+cot^2θ= | cosec^2θ |
| sin(A+B)= | sinAcosB+cosAsinB |
| sin(A-B)= | sinAcosB-cosAsinB |
| cos(A+B)= | cosAcosB-sinAsinB |
| cos(A-B)= | cosAcosB+sinAsinB |
| tan(A+B)= | (tanA+tanB)/(1-tanAtanB) |
| tan(A-B)= | (tanA-tanB)/(1+tanAtanB) |
| sin2A= | 2sinAcosA |
| cos2A= | cos^2(A)-sin^2(A)=2cos^2(A)-1=1-2sin^2(A) |
| tan2A= | 2tanA/(1-tan^2(A)) |
| Put 3sinx+4cosx in the form Rsin(x+α) | Rsin(x+α)=Rsinxcosα+Rcosxsinα Rcosα=3 Rsinα=4 R^2=25 R=5 tanα=4/3 α=arctan(4/3) α=53.1 5sin(x+53.1) |
| How do you convert from parametric form to cartesian form | -Find t in terms of x -Substitute this into the y equation |
| How do you sketch a parametric curve | -Draw a table -Plot points -Sketch curve |
| The domain of f(x) is.... The range of f(x) is.... (parametric equations) | The domain of f(x) is the range of p(t) The range of f(x) is the range of q(t) |
| y=sinkx dy/dx= | kcoskx |
| y=coskx dy/dx= | -ksinkx |
| y=e^kx dy/dx= | ke^kx |
| y=lnx dy/dx= | 1/x |
| y=a^kx dy/dx= | (a^kx)klnx |
| What is the chain rule | dy/dx=(dy/du)x(du/dx) |
| What is the product rule | If y=uv dy/dx=u(dv/dx)+v(du/dx) |
| What is the quotient rule | If y=u/v dy/dx=(v(du/dx)-u(dv/dx))/v^2 |
| y=tankx dy/dx= | ksec^2(kx) |
| y=coseckx dy/dx= | -k(coseckx)(cotkx) |
| y=seckx dy/dx= | k(seckx)(tankx) |
| y=cotkx dy/dx= | -kcosec^2(kx) |
| y=arcsinx dy/dx= | 1/√(1-x^2) |
| y=arccosx dy/dx= | -1/√(1-x^2) |
| y=arctanx dy/dx= | 1/(1+x^2) |
| (d/dx)f(y)= | f'(y)dy/dx |
| The function f(x) is concave if f''(x)...... | f''(x)≤0 |
| The function f(x) is convex if f''(x)...... | f''(x)≥0 |
| How do we find a point of inflection | f''(x)=0 |
| How can you show that there is a root in an interval | The function changes sign |
| How do you solve a function using iteration | Rearrange the function into the form x=g(x), and use the formula x(n+1)=g(x(n)) |
| What is the Newton-Raphson formula | x(n+1)=x(n)=(f(x(n))/f'(x(n))) |
| ∫x^n dx= | (x^(n+1))/(n+1)+c |
| ∫e^x dx= | e^x+c |
| ∫(1/x) dx= | lnIxI+c |
| ∫cosx dx= | sinx+c |
| ∫sinx dx= | -cosx+c |
| ∫sec^2(x) dx= | tanx+c |
| ∫(cosecx)(cotx) dx= | -cosecx+c |
| ∫cosec^2(x) dx= | -cotx+c |
| ∫(secx)(tanx) dx= | secx+c |
| f'(ax+b) dx= | (1/a)f(ax+b)+c |
| How do you integrate expressions in the form ∫k(f'(x)/f(x) | Try y=lnIf(x)I differentiate it and make any adjustments |
| How do you integrate expressions in the form ∫k(f'(x)(f(x))^n | Try y=(f(x))^n+1 |
| How do you integrate by substitution | -Set u= -Find du/dx -Rearrange the u formula to find x -Substitute -Simplify -Integrate -Simplify |
| ∫u(dv/dx)= | uv-∫v(du/dx) dx |
| b ∫(f(x)-g(x) dx= a | b--------b ∫f(x) dx-∫g(x) dx a--------a |
| b ∫y dx≈ a | (1/2)h(y0+2(y1+y2...+y(n+1))+yn |
Created by:
Liam P
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