Semester 1 Exam Flashcards
(97 cards)
1
Q
a^1/2 =
A
n sqrt a
2
Q
a^m/n =
A
(a^n)^m
3
Q
12^n x 18^-2n
A
1/3^3n or (1/27)^n
4
Q
Other ways to write 6^n
A
6(6^n-1)
5
Q
Other ways to write 3^n+1
A
3^2 x 3^n-1
6
Q
Draw the y=a^x graph
A
7
Q
log a n = x =
A
a^x=n
8
Q
log a (a) =
A
a
9
Q
What’s an important note when dealing with inequalities?
A
When dividing or multiplying by a negative, the sign (e.g >) flips the other way
10
Q
Solve for x: 2^-2x+1 < 1/8
A
x>2
11
Q
log a (1/n) =
A
-log a n
12
Q
log a (m^p) =
A
p log a m
13
Q
Solve 2^2x-3 = 47 using logs of base 10
A
x=1/2 (log10(47)/log10(2) + 3)
14
Q
Solve 2^2x-3 = 47
A
x = (log2(47)+3)/2
15
Q
Solve 2 log10 (x) - log10(x+3) = log 10(1/2)
A
x=3/2
16
Q
Solve log2 (2x+1) - log2 (x-1) = 4
A
x=17/14
17
Q
The log of a number less than one is …. hence …
A
It is negative. Hence when dividing/multiplying it make sure to flip sign of inequality.
18
Q
Domain of f(x) =
A
Range of f’(x)
19
Q
Domain of f’(x) =
A
Range of f(x)
20
Q
Solve 0.8^x < 0.4
A
x> log10(0.4)/log10(0.8)
21
Q
Log functions are the inverse of…
A
Exponentials
22
Q
What do we do to find the inverse of something and what do we write in the answer box?
A
Swap x and y.
Remember to write f-1(x) =
23
Q
Draw the base graph for y = log a x
A
24
Q
a^3 - b^3 =
A
(a-b)(a^2+b^2+ab)
25
a^3+b^3 =
(a+b)(a^2+b^2-ab)
26
What does remainder theorem state?
To find the remainder where a polynomial P(x) is divided by (x-a) just evaluate P(a)
27
If there is 2 repeated x-ints what does it mean?
It's a turning point, hence changes direction
28
If there is 3 repeated x-ints what does it mean?
It is a stationary point of inflection
29
Is the gradient 0 at point of inflections?
No
30
Is the gradient 0 at stationary point of inflections?
Yes
31
Draw and tell me about the graph f(x) = a(x-h)^3 + k
This represents a cubic! The stationary point of inflection is (h,K)
32
Draw a quartic graph
33
If I'm asked to state the quotient, what do I need to say
Just what you produce on top of long division, NO need for remainder
34
m= tan of theta or otherwise written as to find theta
theta = tan^-1(m)
35
RATE =
DERIVERATIVE
36
GRADIENT =
DERIVERATIVE
37
What's another way to write f''(x)?
d^2y/dx^2
38
If f'(x) > 0 , then
it is a local minimum
39
If f'(x) < 0, then
it is a local maximum
40
If f'(x) = 0, then
It may be a point of inflection
41
How do you find turning points for quadratics only?
x= -b/2a
42
What is the quadratic formula?
x=-b +- sqrt b^2-4ac / 2a
43
What is absolute minimum and absolute maximum and how do you write it?
It refers to the higher or lowest point that the function reaches in a given interval. GIVE JUST THE Y-VALUE (e.g y=....)
44
How do you actually find out what the absolute minimum or maximum is by hand?
First check 'critical points' (where f'(x) = 0 as these are where slope is 0)
Also check domain x-values by subbing into original eqn to find corresponding y-value.
Whichever has the highest/lowest y-value is the absolute min/max
45
What does it mean for a function to be continuous?
A function is said to be continuous if it can be drawn without picking up your pencil.
46
What is the little rule used to show continuity/discontinuity? And what does 'a' mean?
A function is defined at x=a
It is continuous if lim x>a f(x) = f(a)
47
What's another word for a piecewise function?
Hybrid
48
Say if/where the discontinuty is for the function f(x) = 2x, x greater than or equal to 0 and then -2x+1, x < 0
Well, since 0 is defined for the first eqn, I solve for f(0) = 2(0) = 0
Since 0 is not defined for the second eqn, I have to set up a limit. lim x>0 -2(0) + 1 = 1
Since f(a) doesn't equal lim x>a then at a, which is 0, there is a discontinuity
49
End points are/aren't differentiable
AREN'T
50
If a function is differentiable at a point, it is/isn't continuous
IS
51
sin(theta)^2 + cos(theta)^2 =
1
52
Note for working with the eqn; sin(theta)^2 + cos(theta)^2 =1
Take account of domain. For example, if finding cos(theta) and the domain is pi/2
53
If sin(theta) = 1/2, find the value of cos(theta) for pi/2
-sqrt3/2
54
Draw the y=sin(x) graph and state the min, max, median, amplitude, period and y-int
max = 1
min = -1
median = 0
amplitude = 1
period = 2 pi
y -int = (0,0)
55
Draw the y=cos(x) graph and state the min, max, median, amplitude, period and y-int
max = 1
min = -1
median = 0
period = 2 pi
amplitude = 1
y - int = (0,1)
56
What does A mean and what is it usually?
It represents amplitude. It is positive unless there has been a reflection in the horizontal (x-axis)
57
sqrt 2 =
1.4
58
sqrt 3 =
1.7
59
2sqrt3 =
3.5
60
How do you calculate the period for a cosine or sine graph?
Period = 2pi/n
61
How do you calculate the period for a tan graph?
Period = pi/n
62
What are the steps in solving trig eqns?
1. Rewrite eqn
2. Find RA
3. For rewritten eqn, find what quadrant the answer lies in like basically if it's cos and answer is positive it will be in the A and C quadrant
4. Adjust/rewrite the domain
5. Write down the first lap solutions
6. + or - 2pi until not in the domain anymore and cross out
63
How do you calculate x-ints for a tan graph? (no vertical translations)
x=kpi/n where k is any number (e.g -2, -1, 0, 1, 2)
64
How do you calculate x-ints for a tan graph? (with vertical translations)
Have to solve trig eqn like you would do with a cosine and sine graph
65
Equation for assymptotes on a tan graph
(2k+1)pi/2n
66
What's the formula for another coordinate you could include on a tan graph if there is dilations?
(pi/4n, a)
67
Draw the base graph for a tan graph
68
Explain what to do for inverse of sin (a)
if a>0, answer is RA
if a < 0, answer is -RA
69
Explain what to do for inverse of cos (a)
if a>0, answer is RA
if a < 0, answer is pi-RA
70
Explain what to do for inverse of tan (a)
if a>0, answer is RA
if a < 0, answer is -RA
71
tan (theta) =
sin (theta) / cos (theta)
72
(x, y) in terms of cos and sin
(cos, sin)
73
0/1 =
0
74
1/0 =
Undefined
75
Eqn to convert degrees to radians
x pi/180
76
Eqn to convert radians to degrees
x 180/pi
77
What is a unit that COULD be used for radians?
^c
78
How to find sin/cos/tan of (...)
1. Find RA
2. What quadrant does it lie in
3. Apply the positive or negative sign to the reference angle found
79
sin (3pi/2+theta) =
-cos(theta)
80
cos(3pi/2+theta) =
sin(theta)
81
sin(3pi/2 - theta) =
-cos(theta)
82
cos(3pi/2 - theta) =
-sin(theta)
83
sin(pi/2+theta) =
cos(theta)
84
cos(pi/2+theta) =
-sin(theta)
85
sin(pi/2-theta) =
cos(theta)
86
cos(pi/2-theta) =
sin(theta)
87
Formula to calculate amplitude
max-min/2
88
Formula to calculate median
max+min/2
89
Sin(-7pi/4)
root 2 /2
90
Solve cos(x/2) = -sqrt 3 sin (x/2) for xE 0 to 6pi inclusive (L4 CIRC REVISION BKLT)
x=5pi/3, 11pi/3, 17pi/3
91
Sketch the graph of y=tan(x/2) + 1 for -2pi to 2pi inclusive
PAGE 4 OF L4 CIRC FUNC REVISION BKLT
y-int should be (0,1), end-points should be (-2pi,1) and (2pi,1), x-ints include -pi/2, 3/2pi...
92
CAN USE CAS; 3x-y=18 and x is positive then the minimum value of x^2y is ...
-96
93
Write 8x^3 - 27 as a product of a linear and quadratic factor
(2x-3)(4x^2+6x+9)
94
Solve the following polynomial for x; 2x^3 +x^2 - 27x - 36 = 0
x = -3/2, 4, -3
95
Sketch (4x-2)^3 - 3
IN LESSON 2 REVISION BKLT
y-int is at (0,11), x-int is (little 3 sqrt 3+2 / 4 , 0) and turning point is (1/2, -3)
96
The function f(x) = x^3 undergoes the following transformations in the order given:
Reflection in x-axis
A dilation of factor 1/3 from the y-axis
A translation of 1 unit in the negative direction of the x-axis
The transformed function is expressed by g(x) = -(ax+b)^3 + c
Find the values of a, b and c
a= 3
b= 3
c = 0
97
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