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Gravity
Key facts useful to know for the Edexcel C1/C2 exams.
Terms in this set (104)
a^m x aⁿ =
What is a to the power of minus m?
Factorise x² - 49
(x - 7)(x + 7)
What is the quadratic formula?
where a is the coefficient of x², b is the coefficient of x and c is the constant.
How do you rationalise a denominator?
Multiply both numerator and denominator by the conjugate
What should you do when simplifying a surd?
Look for the largest square number that is a factor, then split it up.
What is a to the power of zero?
1
How do you find where a graph crosses the x-axis?
Let y=0
How do you find where a graph crosses the y-axis?
Let x=0
What shape is a x squared graph?
u (hand signal)
What shape is a minus x squared graph?
n (hand signal)
How do you complete the square?
1) Half the coefficient of x and put it in the bracket with x
2) subtract that number squared
3) put the constant on the end.
Complete the square for x² + 8x + 2
(x + 4)² - 16 + 2
(x + 4)² - 14
What is the discriminant?
b² - 4ac
When are there two unique real roots?
When b² - 4ac > 0
When are there no real roots?
When b² - 4ac < 0
When is there two equal roots?
When b² - 4ac = 0
What happens when you multiply/divide an inequality by a negative?
The sign switches
How do you find where a line and a curve meet?
If they are both in y= format, put them equal to each other.
If not, try substitution.
How do you solve a quadratic inequality?
Sketch the graph then shade in the region(s) of the curve that work.
Where does y=(x-2)(x-1)(x+3) cross the x-axis?
(2,0) (1,0) and (-3,0)
What shape is the graph y=x³+2x²+x+1
Hand gesture up, down and up
What does y = 1/x look like?
Where are the asymptotes for the y=1/x graph?
x=0 (y-axis)
y=0 (x-axis)
What does y=-1/x look like?
Same shape as y=1/x but in the other quadrants.
What effect does -f(x) have on a graph f(x)?
reflection in the x-axis
What effect does f(x) + 3 have on a graph f(x)?
Moves the graph up 3 squares (adds 3 to all the y-coordinates)
What effect does 2f(x) have on a graph f(x)?
Stretches the graph by scale factor 2 in the vertical direction. (Doubles all the y-coordinates)
What effect does f(-x) have on a graph f(x)?
Reflection in the y-axis
What effect does f(x+2) have on a graph f(x)?
Moves the graph 2 squares to the left. (Subtract 2 from all the x-coordinates)
What effect does f(2x) have on a graph f(x)?
Compresses it by scale factor 2 in the horizontal direction. (Halves all the x-coordinates)
What is the equation of a straight line?
y=mx + c, where m is the gradient and c is the y-intercept.
Or ax + by + c = 0 and a,b,c are all integers.
How do you find the gradient between two points?
rise over run
How do you find the equation of a line, if you know the gradient and a point it passes through?
Or
Use y=mx+c but substitute in the number for the gradient and the coordinate for x and y to work out the +c.
Two lines are parallel, so their gradients are....
the same
A line has gradient m and a line is perpendicular to it. It's gradient is?
How do you prove Sn for an arithmetic series?
1) Write out Sn as Sn = a + (a + d) + (a + 2d) ... etc
2) Reverse it.
3) Add the two lines together.
4) Make all the terms inside the bracket 2a + (n-1)d
5) Since there are n of them, write n(2a + (n-1)d)
6) Divide by 2.
What is Sn for an arithmetic series?
What is Un for an arithmetic series?
How do you differentiate?
Multiply by the power, then reduce it by 1
What do you get if you differentiate numbers?
zero.
What do you get if you differentiate x?
1
Differentiate 5x⁴
20x³
How do you find the equation of a tangent to a curve at a point (x,y)?
1) Differentiate the equation of the curve.
2) Sub in the x coordinate into dy/dx to find the gradient. This is the m in your y=mx+c
3) Sub in the coordinate into y=mx+c to find the c.
Done
How do you find the equation of a normal to a curve at a point (x,y)?
1) Differentiate the equation of the curve.
2) Sub in the x coordinate into dy/dx.
3) Take the negative reciprocal of this and this is the m in your y=mx+c
4) Sub in the coordinate into y=mx+c to find the c.
Done
How do you integrate?
Increase the power of x by 1, and divide by the new power.
What do you get if you integrate 4?
4x + c
What should you never forget when integrating?
+ c
Integrate 6x²
2x³ + c
You have been given dy/dx and a point on a curve. How do you find it's equation?
1) Integrate dy/dx
2) Sub in the coordinate to find c.
How do you show (x - a) is a factor of f(x)
Sub in a and show it equals 0.
THEN WRITE, THEREFORE IT IS A FACTOR.
How do you find the remainder when dividing by (x - a)?
Sub in a, and the numerical answer is the remainder.
What is factor theorem?
(x - a) is a factor of f(x), then f(a) = 0
What is remainder theorem?
If dividing f(x) by (x-a), then f(a) is the remainder.
What is the Sine rule?
What is the Cosine rule?
What is the area of a triangle?
or 1/2base x height
Write a^x = n as a logarithm
What is log (to base 4) of 16?
2
What is log (to base a) of a?
1
What is the log of 1?
0
How do you solve 6ⁿ = 11
Take logs of both sides and solve.
Log x + log y =
Log xy
Log xⁿ
nlogx
Log x - Log y =
Log x/y
How do you solve 6²ⁿ + 3(6ⁿ) - 10 = 0?
Rearrange to (6ⁿ)² + 3(6ⁿ) - 10 then treat as a quadratic.
How do you find the midpoint between two coordinates?
Add together then half.
How do you find the distance between two points?
Pythagoras
What is the equation of a circle?
Where (a,b) is the centre and r is the radius.
How do you find the equation of a tangent to a circle, at a given point?
1) Find the gradient of the radius.
2) Take the negative reciprocal
3) Write as y=mx + c
4) Sub in the coordinate to find c.
How do you calculate ⁶C₂?
Use your calculator (button above divide)
How do I use the binomial expansion on (a + bx)ⁿ?
1) Write (n0) then aⁿ
2) The next term is (n1)aⁿ⁻¹(bx)¹
3) The next term is (n2)aⁿ⁻²(bx)²
and so on...
What is π radians in degrees?
180
What is 360 degrees in radians?
2π
What is 90 degrees in radians?
π/2
What is 60 degrees in radians?
π/3
How do you convert from degrees to radians?
1) Divide by 180
2) Times by π
How do you convert from radians to degrees?
1) Divide by π
2) Times by 180
How do you work out the arc length?
l = rθ
That means times the radius by the size of the angle (in radians)
How do you find the area of a sector?
A = ½r²θ
Where r is the radius and θ is the angle.
How do you find the area of a triangle in radians?
It's still ½abSinC, but make sure you are in radians mode.
How do you find the common ratio of a geometric series?
Divide a term by the previous term.
or if you knew 5th and 3rd, you would do ar⁴÷ar² to find r² then square root.
What does a geometric sequence look like?
What is Un for a geometric series?
arⁿ⁻¹
What is Sn for a geometric series?
How do you prove Sn for a geometric series?
What is the sum to infinity for a geometric series?
When will a geometric series converge?
When -1<r<1
What does the CAST diagram look like?
What does the Sine Graph look like?
What does the Cosine Graph look like?
What does the Tangent Graph look like?
Asymptotes at x=90, x=270 etc
What does the y = 2^x graph look like?
How do you find a maximum/minimum/turning point/stationary point?
1) Differentiate the equation of the curve to find dy/dx
2) Put dy/dx = 0 and solve to find x.
3) Put the x value back into the equation of the curve.
How do you find d2y/dx2?
Differentiate y=.... once to find dy/dx and then differentiate again to find d2y/dx2
At a minimum, what will happen to the second derivative?
At a maximum, what will happen to the second derivative?
Tanθ =
State an identity linking sin²θ and cos²θ
sin²θ + cos²θ = 1
What are the steps in solving sinθ=0.5 between 0≤θ≤360
1) Find sin⁻¹0.5 .... 30°
2) Draw the CAST diagram
3) Since 0.5 was positive, it is in the A and S quadrants.
4) Start at 0 and go around anti-clockwise.
That will give 30 and 150.
What are the steps in solving 5sinx = -2 between 0 and 360.
Rearranging gives Sinx = -2/5
1) Find sin⁻¹(2/5) .... 23.6 (ignore -)
2) Draw the CAST diagram
3) Since it was -2/5, then we go for quadrants sin is negative in, C and T.
4) Start at 0 and go around anti-clockwise.
5) 203.6 and 336.4
How do you find the area under and curve, between x=a and x=b?
Integrate the curve between b and a.
That means integrate, sub in b and a and subtract.
What will happen if the area between the curve and x-axis is below the x-axis?
It will be a negative answer when integrated... just make positive.
What is the trapezium rule?
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