Study Material (Maths)

(first point is at π/3, so when sin(2x + π/3) = 0)

Find the area under the graph of y = sin(2x + π/3) from x = 0 as far as the first point at which the graph cuts the positive x-axis.

(c = 1/(b - a)*∫f(x) dx (intergral is from a to b)) 22/3

Find the average value of the function f(x) = x^2 + x + 1 from 1 to 3.

(-1, -4) local minimum, (1, 4) local maximum

Find the coordinates of the stationary points of the curve y = 6x - 2x³ and determine their nature.

(if y = f(x)g(x), then dy/dx = f'(x)g(x) + f(x)g'(x)) sin(x)^4*cos(x)^2*(5cos(x)^2-3sin(x)^2)

Find the derivative with respect to x of sin^5 x cos^3 x

(if y = f(x)g(x), then dy/dx = f'(x)g(x) + f(x)g'(x)) 3x^2 sinx + x^3 cosx

Find the derivative with respect to x of x^3 sin x

(if y = f(x)g(x), then dy/dx = f'(x)g(x) + f(x)g'(x)) exp(3x)*(3x+1)

Find the derivative with respect to x of xe^(3x)

(f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x))/[v(x)]^2) (2x*cos(x)+x^2*sin(x))/cos(x)^2

Find the first derivative of f(x) = x^2/cos x

g⁻¹(x) = 2/x - 5, x ∈ ℝ, x ≠ 0

Find the inverse of g(x) = 2/(x + 5), x ∈ ℝ, x ≠ -5

(S = ∫√(1 + (f'(x))^2) dx) 1

Find the length of f(x) = 2 between x=2 and x=3

(S = ∫√(1 + (f'(x))^2) dx) 9.073

Find the length of y = x^(3/2) from x = 0 to x = 4.

(3 sin x + 2 cos x = R sin(x + α), 3 = R cos α and 2 = R sin α, α = 0.588 rad, R = √13 therefor 3 sin x + 2 cos x = √13 sin(x + 0.588). The max and min values of this function is ±√13, the max/min values are achieved when sin(x + 0.58..) = 1 or -1, so just solve for x (you may get -2.1 as the answer for one but it has to be positive.) x = 0.99, 4.12

Find the maximum and minimum values of 3 sin x + 2 cos x and find in radians to two decimal places the smallest positive values of x at which they occur.

(Domain is ℝ, but you can consider 0 ≤ x ≤ 2π, rule from before, sin(2x) = 2sin(x)cos(x). ) Max/min at x = 0, π, 2π

Find the minima and maxima of f(x) = 4 cos x + cos 2x.

f(x) ≤ 1

Find the range of the function f(x) = x -2, x ≤ 3

h(x) > -4

Find the range of the function h(x) = x³ + 4, x > -2

(When a polynomial p(x) is divided by x - t, let the quotient be q(x) and the remainder be R. Then p(x) = (x - t)q(x) + R) (x + 3 = 0) (x = -3, sub x into the equation) -14

Find the remainder when x³ - 3x + 4 is divided by (x + 3) using the remainder theorem.

x = -2, 4

Find the roots of y=-x^2+2x+8

The Chain Rule

For the equation y = 2(5x + 2)³ would you use the chain rule or the product rule to differentiate?

The Product Rule

For the equation y = x²(5x + 2)³ would you use the chain rule or the product rule to differentiate?

R = √(a^2 + b^2)

Given, a sin x + b cos x = Rsin(x + a), define R in terms of a and b

ff⁻¹(x) = f⁻¹f(x) = x

How do you check that your answer for the inverse f⁻¹(x) is correct?

m1 × m2 = -1

How do you find out that two lines are perpendicular?

A = 4, B = 10, C = 20, R = 43

If 4x³ + 2x² + 3 ≡ (x - 2)(Ax² + Bx + C) + R, find A, B, C and R

The x values

In functions, what is the domain?

The y values

In functions, what is the range?

3x + 1

One factor of 3x² -5x -2 is x - 2. Find the other factor.

(E = mcΔT, E is energy, m is mass, c is specific heat capacity, ΔT is change in temp, E = mL, E is energy, m is mass, L is specific latent heat, 0 C = 273.15 K) 23.543 C

QUESTION A 100 g sample of gold at a temperature of 260 ℃ is added to 500 g of water at 25 ℃. What is the final temperature of the gold sample and of the water? Assume a constant specific heat of water and gold of 4.187 kJ/kgK and 0.129 kJ/kgK respectively. Provide you answer with at least 3 decimals.

(E = mcΔT, E is energy, m is mass, c is specific heat capacity, ΔT is change in temp, E = mL, E is energy, m is mass, L is specific latent heat) 0.430 kg

QUESTION A sample of water absorbs 90 kJ of energy when it is heated from 40 C to 90 C. What is the mass of the water sample? Assume a constant specific heat of water of 4.187 kJ/kgK. Provide you answer with at least 3 decimals.

0.00833 m

QUESTION A wave travels with a velocity of 2.5m/s and a frequency of 300Hz. What is its wavelength?

(when the angle in the sine is divided by 2, the graph is stretched horizontally by a factor of 2 and vertically by a factor of 2 squared (4), you also need to shift both functions independently of each other - theory part - if the angle is divided by three, the graph will stretch horizontally by a factor of 3 and stretched vertically by a factor of 3 squared, so 9. - The shift is true for all trig functions) g(x) = (x+1)^2*sin(x+1)

QUESTION Consider the graph of f(x)=x^2 sin(2 x). We want to define a new function g(x) by performing the following transformations: 1. Stretching the graph horizontally by a factor 2 and vertically by a factor 4 2. Shifting the graph to the left by a value 1. What is g(x)?

4/x

QUESTION Determine the derivative of f(x) = 4 ln(x).

4/cos(4*x)^2

QUESTION Determine the derivative of f(x) = tan(4x).

-cos(4x+2)/4 + C

QUESTION Evaluate the following integral ∫sin(4x+2)dx.

a = 7.586 m/s^2, E = 484000 J, F = 676.9231

QUESTION In function of a crash test, a car starts from rest and crashes against an immovable wall 2.9s later with a velocity of 22m/s. During this crash the frontal structure of the car absorbs the forces exerted on the car by the wall. The car has a total mass of 2000kg. What was the acceleration "a" of the car? Provide your answer with at least 3 significant figures. Compute the kinetic energy "E" of the car just before it hits the wall. Round your answer to the nearest whole number. The potential energy absorbed by the frontal crash structure is given by Ep=F⋅s. (i.e. the product of the average force exerted on the structure and the derformation distance of the structure) The total length of the crash structure of the car, designed to absorb the impact, is 1.1m. What is the average force "F" in kN exerted on the crash structure when 65% of the crash structure deforms (note the unit!)? Provide your answer with at least 4 significant figures.

l = 3.722 m, a = 14.51 m^2

QUESTION The surface area of an equilateral triangle△ABC is equal to 6m^2. State the length of the sides "l" and the area of the circle "a" encapsulating the triangle. (4 sig figures)

t = 4.89 s, f = 0.204 Hz

QUESTION What is the period "t" of a wave which has a velocity of 2.35m/s and a wavelength of 11.5m? What is the frequency "f" of this wave? (3 sig figures)

(V = ∫πy²dx (with b at the top of ∫ and a at the bottom) about the x-axis, where a and b cross the x axis) V = 78096*Pi/7

QUESTION f(x) = x^3, Let A be the area enclosed by f(x) and the lines x=5 and y=1, as shown in the figure (not drawn to scale - not existing lol). Determine the volume V of the body of revolution that is created by rotating A over the x-axis. Give an exact answer.

3-2*exp(x)

QUESTION ln(2/(3-y) = -x. Determine y as a function of x, simplify your answer.

(Choose a random point P, (x, sin x) so sinx/x is the gradient. As x tends to 0, this tends to the gradient of the origen. Using the rule where as x tends to zero for sin x/x, it equals 1. So the gradient is 1.)

Show that the graph of y = sin x has a gradient of 1 at the origin (not using differentiation)

Maximum at (0.5, 4) ( (x + p)² + q and max/min coords at (-p, q) )

Sketch and state the coordinates of the vertex and wheather it is a maximum or a minimum for the following equation. y = 3 + 4x - 4x²

Minimum at (-1, -1) ( (x + p)² + q and max/min coords at (-p, q) )

Sketch and state the coordinates of the vertex and wheather it is a maximum or a minimum for the following equation. y = x² + 2x

1/2

Sketch the graph of y = xe^(-x^2) for x > 0. Find the area contained between this graph and the positive x axis.

(if h = 2, then area = 5/6 for trapezium rule) (exact answer is ln(9/4)) ln(9/4)

Sketch y = 2 / (x + 1) and use your sketch to make a rough estimate of the area under the graph between x = 3 and x = 5. Compare your answer with the exact answer.

(convert equation in form of Rsin(x + a) = √13sin(x + 0.588), and then solve) x = -0.31, 2.27

Solve the equation 3 sin x + 2 cos x = 1, for -π ≤ x ≤ π to 2 decimal places

y = 3x² - 2x

The gradient of a curve is given by dy/dx = ax + b. Given that the curve passes through (0, 0), (1, 1) and (-2, 16), find the equation of the curve

(subtract the areas of each of the graphs) 3/2 - 2 ln2

The graphs of y = 2 / (x - 2) and y = -x - 1 intersect where x = 0 and x = 1. Find the area of the region between them.

(when the gradient is max - rising fastest - h' is equal to 0.5 (as 0.5 is the max point on that graph and that means the highest rate of change as h' is the gradient) therefore t = 6sin⁻¹(-1)/π, 6(sin⁻¹(-1) + 2π)/π, 6(sin⁻¹(-1) + 4π)/π, etc) 9 am and 9 pm

The height in metres of the water in a harbour is given approximately by the formula h = 6 + 3 cos(πt/6), where t is the time measured in hours from noon. Find an expression for the rate at which the water is rising at time t. When is it rising the fastest?

p = -4, q = -5

The mid-point of the line joining the points A(p, -2) and B(6, -8) is (1, q). Find p and q.

P(-3.5, 3.5)

The normals at the points (0, 0) and (1, 2) to the curve y = x + x³ meet at point P. Find the coordinates of point P.

(f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x))/[v(x)]^2) sec(x)^2

Using the quotient rule, find d/dx of tanx

1. Let f(x) = y. 2. Change x to y and y to x. 3. Rearrange to get x in terms of y 4. Write in the correct format

What are the steps used to find the inverse function f⁻¹(x) given f(x)?

h(a + b)/2

What is the area of a trapezium?

A = 0.5 × ab sin(C)

What is the area of a triangle

c = 1/(b - a)*∫f(x) dx (intergral is from a to b)

What is the average value "c" of a function f(x)

If y is a function of u, and u is a function of x, dy/dx = dy/du × du/dx

What is the chain rule?

2πr

What is the circumference of a circle?

(x-h)^2 + (y-k)^2 = r^2 (where h and k are the coordinates of the centre of the circle.)

What is the equation for a circle?

(for general equation y = ax^2 + bx + c) x = -b/2a

What is the equation for axis of symmetry for a quadratic?

f(x) = ab^x (a = initial value when x = 0) (b = constant which indicates how fast quantity is growing) (x usually represents time)

What is the equation for continuous exponential growth?

V = ∫πy²dx (with b at the top of ∫ and a at the bottom, where a and b are lines of x)

What is the equation for the volume of a solid revolution that is generated by rotating the curve y = f(x) between x = a and x = b through 360° about the x-axis?

V = ∫πx²dy (with b at the top of ∫ and a at the bottom, where a and b are lines of y)

What is the equation for the volume of a solid revolution that is generated by rotating the curve y = f(x) between y = a and y = b through 360° about the y-axis?

(x1 + x2) / 2, (y1 + y2) / 2

What is the equation of the midpoint of a line?

S = ∫√(1 + (f'(x))^2) dx

What is the formula for arc length of f(x).

a^2 = b^2 + c^2 - 2bcCosA

What is the law of Cosines.

sin A/a = sin B/b = sin C/c (where A is the angle and a is the length)

What is the law of Sines.

(-2.5, 1.5)

What is the midpoint of the straight line joining the sets of points (-8, 5) and (3, -2)

If y = f(x)g(x), then dy/dx = f'(x)g(x) + f(x)g'(x)

What is the product rule?

If f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x))/[v(x)]^2

What is the quotient rule?

When a polynomial p(x) is divided by x - t, let the quotient be q(x) and the remainder be R. Then p(x) = (x - t)q(x) + R

What is the remainder theorem?

A = 1/2 × r²θ

What is the sector area A of a circle?

1

What is the value of (log base b) of b.

V=π*r^2*h/3

What is the volume of a cone?

V=lwh/3 (l = base length) (w = base width) (h = height)

What is the volume of a pyramid?

The domain of the inverse will take the value of the range from the original function

When finding the inverse of a function what value will the domain of inverse take?

The range of the inverse will take the value of the domain from the original function

When finding the inverse of a function what value will the range of inverse take?

(3 1)

Write down the displacement from A(2, 6) to B(5, 7) as a column vector.

(Equate the two expressions 3 sin x + 2 cos x = R sin(x + α), 3 = R cos α and 2 = R sin α, α = 0.588 rad, R = √13, the max/min values occur at π/2 and 3π/2 ) y = √13sin(x + 0.588)

Write y = 3 sin x + 2 cos x in the form y = R sin(x + α)

(you replace any value of x by 1/a) y = f(x/a)

y = f(x), state the new equation for y if you want to stretch it in the x axis by a factor of a.

y = af(x)

y = f(x), state the new equation for y if you want to stretch it in the y axis by a factor of a.

18 m^3

A closed box with a square base has a total surface area of 36 m². Find the maximum possible volume of the box.

y = 3x - x³

A curve is such that d²y/dx² = -6x and the curve has a maximum point at (1, 2). Find the equation of the curve.

(3, 0)

A diagram with a straight line AB and a perpendicular line PC. The point A is (-3, -4), the point B is (6, 2) and the point C is (5, -3). Find the coordinates for point P.

64 cm²

A rectangle has a width of x cm. The perimeter of the rectangle is 32 cm. Find the maximum area of the rectangle

24.5 cm²

A right angled triangle has a width of x cm. The length of the hypotenuse is 10 cm. The perimeter of the triangle is 24 cm. Find the maximum area of the triangle.

3i + j - 2k

Convert the column vector (3 1 -2) into unit vector form.

xi + yj

Convert the column vector (x y) into unit vector form.

A quantity which has both magnitude (size) and direction.

Define a vector

(first step is to find the axis of symmetry. For general equation y = ax^2 + bx + c, the axis of symmetry is at x = -b/2a. axis of symmetry is at x = 2, just move either side of the line x = 2 to find two symmetrical points. For two units to either side of x = 2. x = 0 and 4, sub in these values to initial equation and you get y = 7.) (4, 7) and (0, 7)

Determine a pair of symmetric points for y = x^2 - 4x + 7. Explain your thinking.

( easy way to think of it (cos x)⁴ ) -4 sin x cos³ x

Differentiate cos⁴(x), with respect to x.

(d/dx (cos x)⁻¹ ) sin x/ cos² x

Differentiate sec x, with respect to x.

3 cos(3x - π/4)

Differentiate sin(3x - π/4), with respect to x.

(2 9)

Evaluate the vector (3 4) + (-1 5)

(2 6)

Evaluate the vector (3 4) - (-1 5)

(find value of R in the form Rsin(x-a), R is the max range of y values (for x values its called the domain) ) 5 is bigger than √17 and therefore there are no solutions.

Explain why the equation sin x° - 4 cos x° = 5 has no solutions

(differential of function over the original function, )d/dx sec x)/sec x = sec x tan x / sec x) tan x

Find d/dx ln(sec x)

-4

Find log₁/₃ 81 without a calculator.

4

Find log₃ 81 without a calculator.

7.2 cm

Find the arc length of a circle with a 12 cm radius and an angle of 0.6 rad

There is no answer as the area -2 < x 4 doesn't exist.

Find the area under the graph of y = 1/x from x = -2 to x = 4.

(= [ln x]⁴₂) (check the answer using the trapezium rule, it should be smaller than the trapezium rule.)

Find the area under the graph of y = 1/x from x = 2 to x = 2 to x = 4.

(Divide by the differential of the power) 0.5(e² - 1)

Find the area under the graph of y = e^2x from x = 0 to x = 1