Study Flashcards
Why is 3ln(1/2) < 2ln(1/2)?
3ln(1/2) = 3ln(2^-1)=-3ln(2) and 2ln(1/2)=2ln(2^-1)=-2ln(2).
-3ln(2) < -2ln(2)
If a function has a curved top what is the interval?
Closed
An interval is closed when asked to find where a function is increasing/decreasing when
The top is rounded
Why is sin^-1(sin 4π/3) = -π/3?
sin(π-θ)= sin, so:
sin(π-4π/3)=sin(-π/3) which is in the range of sin^-1.
sin(π-θ)= sin, so:
sin(π-4π/3)=sin(-π/3) which is in the range of sin^-1.
sin^-1(sin 4π/3) = -π/3
What does it mean to find a domain where f is one-to-one?
Find where it is defined
Simplify with exponent rules:
√a √b/ √ab
a^1/6 / b^1/12
Find domain:
h(x)=1/ ^4√x^2 -5x
(-∞, 0) U (5, ∞) because x(x-5)>0 is equal to x<0 and x>5
Find domain:
f(x)= √16-x^4 / x^2 -1
[-2,-1)U(-1,1)U(1,2] because x cannot equal +-1 and x<=+-2 which equals -2<=x<=2
If a function is always decreasing, never zero, and f(-2)=1 and f(1)=2, what is the graph?
A piecewise function.
What does cos(-x) equal?
cos(x)
What does sin(-x) equal?
-sin(x)
Equals -sin(x)
sin(-x)
Simplify:
2x(4-x)^-1/2 - 3√4-x = 0
x=12/5 because you put (4-x)^-1/2 under 2x and ass 3√4-x to right side. Then solve for x.
How is this absolute value function solved?:
3|x-4|=10
|x-4|=10/3
Then x-4=10/3 and x-4=-10/3
Simplify:
3x^3/2•y^3 / x^2•y^-1/2
9y^7 / x because you multiply all exponents by 2 first. Then any negative exponents are flipped and solve.
How is this simplified?
√200 - √32
Split into one non-squarable and squarable:
√2 • √100 - √2 • √16
√2 • 10 - √2 • 4
6√2
Evaluate:
y/x - x/y / 1/y - 1/x
- (y+x) because (y+x)(y-x) / x-y =
- (y+x)(x-y) / x-y
How is this evaluated?:
16^-3/4
Equals 1/(16^1/4)^3
1/2^3
1/8
How do you prove:
√x^2 = x
√(-2)^2 = -2 so false
How do you prove:
a^2+b^2=c^2
(1)^2+(2)^2=(3)^2 so false
How do you prove:
√xy
Equals √x•√y
x^1/2 • y^1/2
(xy)^1/2 = √xy so true
If the half-life is 4 days, what is the mass after 16 days?
(1/2)^t/4 where t is days and 4 is half-life
Difference between rational and algebraic equations
Rational has polynomial on numerator AND denominator
Rational has polynomial on
numerator AND denominator
Evaluate this log:
log(5)(200/8)
log(5)(25)
so 2
Solve for x:
2^x = 5^3x+7
x=7ln(5) / ln(2) - 3ln(5) because you take exponent out and distribute.
If you have domain and need range, what is done?
Test values of domain for highest and lowest values of range.
Test values of domain for highest and lowest values of range.
If you have domain and need range
Why is cos^-1(-1)= π?
Because cosine hits -1 every π
Because cosine hits -1 every π, this means the inverse would be.
cos^-1(-1)= π
What is the domain of ln(x)<0?
3^ln(x)
Definition of one-to-one funct
Function that never takes on the same value twice
Function that never takes on the same value twice
One-to-one
Find exponential function given two points:
1,12) and (3,108
12=C(b^1)=Cb
108=C(b^3)=Cb^3
C=12/b (solve for C)
108= 12/b•(b^3)=12b^2 (plug in C in second)
b=3 (solve for b)
C=12/3 = 4 so,
y=4(3)^x
How is number e defined?
The value so that the slope of the tangent like to y=e^x at (0,1) is exactly 1.
The value so that the slope of the tangent like to y=e^x at (0,1) is exactly 1.
e
How is this divided?:
9^3 / 3^-8
Flip 1/9^3 / 1/3^8
1/9^3 • 3^8
3^8 / 9^3
3^8 / (3^2)^3
3^8 / 3^6
3^2
The graph of y=1/3x + b is shrunk by a factor of
3 not 1/3
Is f(s+t)=f(s)+f(t)?
No because (f+f)(x)=f(x)+f(x)
Suppose g is an even function and let h=f o g. Is h always an even function?
Yes because if g = x every time being even, then so is f o g
Volume of sphere
4/3πr^3
4/3πr^3
Volume of sphere
Graph the function y=|x^2 -9|
Same as regular x^2 graph but x’s from -3 to 3 are reflected over x-axis
Equation for two families if functions if one family’s is y=5x+b and the other’s is y=m(x-5)+1.
5x+b=m(x-1)+1 and solve
If a box has dimensions 18in. by 30in and gets equal squares of side x at each corner folded up, express the new volume as a function of x.
V=lwh so,
V=(30-2x)(18-2x)x
If the sides of a rectangle box, (not the bases), are the height (h) and the length (2w), and also the width (w), what is the cost of all the sides?
2[2(wh) + 2(2wh)] since the area for 2 sides is 2(wh) total, and the other two sides is 2(2wh) total.
If the material for the base of a rectangular box costs $8 per square meter, what is the total cost if w is the width and 2w is the length?
A=lw so,
(2w)(w)=2w^2
Multiply 8 by this number for the cost:
8(2w^2)
In general, if f(x)=g(x) when x cannot equal a, then
lim f(x) = lim g(x) x—>a x—>a
Substituting limit points directly into functions only works for
Polynomials and rationals
Only works for polynomials and rationals
Substituting limit point directly into a function
Domain of tan function
nπ/2
nπ/2
Domain of tangent function
How is cot graphed?
Tangent but backwards
Tangent but backwards on graph
Cot
Domain and range of cos^-1
R: [0, π]
D: [-1,1]
Domain and range of sin^-1
D: [-1,1]
R: [-2π, 2π]
When reflection about a y value, what is done?
Add negative at base and add twice the y-value
Add negative at base and add twice the y-value
When reflection about a y value
When reflecting about x-values what is done?
Add negative at x and add twice the x-value
Add negative at x and add twice the x-value
When reflecting about x-values
Order of transformations
H-shift H-stretch/shrink Reflect y-axis Vert-stretch/shrink Reflect x-axis Vert-shift
To find x-values of log function what is done?
Set praenthesis equal to 0, 1, and the base and solve
Set praenthesis equal to 0, 1, and the base and solve
To find x-values of log function
