random

Question 1

differentiate a^(kx)

Answer 1

ka^(kx)ln(a)

Question 2

why might the Newton-Raphson method fail?

Answer 2

if f’(x)=0 then the tangent is vertical and values won’t converge (to meet a root)

Question 3

how to integrate parametric equations

Answer 3

int[(y)(dx/dt)]dt

FIND NEW LIMITS (sub in x values to find t limits)

Question 4

differentiate parametric equations

Answer 4

dy/dx = (dy/dt)(dt/dx)

Question 5

how would you split up 30Σr=13 [(-2)^(r)-4r-78]?

Answer 5

• split into:
  30Σr=1 [(-2)^(r)-4r-78] - 12Σr=1 [(-2)^(r)-4r-78]
• split further into arithmetic and geometric sequences:
  30Σr=1 [(-2)^(r)] - 30Σr=1[4r+78]
  and
  12Σr=1 [(-2)^(r)] - 12Σr=1 [4r+78]

Question 6

what is the equation of the line in the MIDDLE of two parallel lines:
5x + 3y = 15
5x + 3y = 83

Answer 6

5x + 3y = 49

Question 7

how can a more accurate approximation be found using the Newton-Raphson method?

Answer 7

use more iterations

Question 8

volume of a sphere formula

Answer 8

V = (4/3)πr³

Question 9

how could you work out the value of k for a function where x≠k?

Answer 9

• usually at an assymptote
• if the function is a fraction, make the denominator equal to zero and solve to find out what value x cannot be
  (if x is this value, the function would be undefined)

Question 10

what is the domain of f-¹(x)?

Answer 10

the range of f(x)

Question 11

C has equation y = x^(x)

find, by taking logarithms, the x coordinate of the turning point on C

Answer 11

lny = xlnx

(1/y)(dy/dx) = lnx + 1

dy/dx = ylnx + y = 0 (there are stationary points at dy/dx = 0)

lnx = -y/y = -1
x = e-¹

Question 12

how would you solve dy/dx = ∞

Answer 12

make the denominator = 0 and solve

(to find a coordinate, solve this equation simultaneously with the original equation)

Question 13

when is a curve concave and convex?

Answer 13

concave: d²y/dx² < 0
convex: d²y/dx² > 0

Question 14

which type of mapping is not a function?

Answer 14

one to many

Question 15

give examples of graph shapes for:
- one to one function
- many to one function

Answer 15

• exponential, logarithmic, linear
• trigonometric, polynomial, cubic

Question 16

give an example of a many to many function

Answer 16

circle

Question 17

one to one function

Answer 17

• x : y values are 1:1
• if you draw a horizontal line and a vertical line they both crossed the graph at one point

Question 18

many to one function

Answer 18

• x : y values are 2:1
• if you draw a horizontal line and it crosses the graph at more than one point
• if you draw a vertical line and it only crosses the graph at one point

Question 19

one to many

Answer 19

• x : y values are 2:1
• if you draw a vertical line and it crosses the graph at multiple points
• if you draw a horizontal line and it crosses the graph at one point

Question 20

which functions have inverses?

Answer 20

one to one functions

Question 21

why don’t many to one functions have an inverse?

Answer 21

the inverse would be a one-to-many which is not a function

Question 22

what is the domain of an inverse function?

Answer 22

range of the original function

Question 23

how can we make a many to one function an inverse?

Answer 23

restrict the domain so it becomes one to one

Question 24

what transformation occurs to inverse functions graphically?

Answer 24

reflection in x=y

Question 25

how would you find an expression for v in terms of t for a = g - 0.1V ?

Answer 25

a = dv/dt so dv/dt = g - 0.1V then separate the variables so int(1/(g-0.1v)dv = int(1)dt solve and find c

Question 26

what happens to the forces on the point of tipping? (moments)

Answer 26

all reaction forces are zero EXCEPT the reaction force from the point it is tipping about