Logarithms
Can you compute log of zero or negative num?
NO
The angle 3π/4 radians in degrees.
135 degree
(replace pi by 180 and multiply).
The hole of the function f(x) = (x2 + 6x - 16)/(x2 - 100).
no holes
(there are no cancellations after factoring, therefore there are no holes in this function).
The limit as x approaches 2 of (3x + 2x2).
14
(basic limit computation)
The derivative of y = x3.
y' = 3x2
(basic power rule)
The x value in 33x = 27-3.
-3
(rewrite with a common base of 3 and distributing the powers to solve for x).
The exact value of sin π/6.
1/2
(convert to degrees and solve using SOHCAHTOA or prior knowledge).
The vertical asymptote of y = (x + 2)/(2x - 10).
x = 5
(vertical asymptotes are found by setting the denominator equal to 0 after simplifying).
The limit as x approaches infinity of 2x/x3.
0
(limit transformations)
The derivative of y = -sqrt(x) - [sqrt(x3)]/3.
y' = -1/2sqrt(x) - sqrt(x)/2
(chain rule but a little more complicated)
The value of x in log2(8x) - 3log2(6) = 1.
27
(use the power rule and then evaluate the exponent).
The value of sec A given that csc A = 13/12.
13/5
(use the pythagorean theorem and plug in the values).
The horizontal asymptote of y = (4x + 3)/(x - 7).
y = 4
(horizontal asymptotes are found by looking at coefficients of the highest exponent when the degree of the numerator is the same as the denominator).
The limit as x approaches infinity of [sqrt(-10 - 5x + 25x6)] / (3 + 4x2).
infinity
(the degree of the numerator is greater than the degree of the denominator)
The f"(x) given the function f(x) = 2x4/3 + x/3 + x-3/2 + x6/6.
f"(x) = 8x2 + 6-5 + 5x4
(higher derivatives)
The expanded form of log (z2x3)/y2.
2 log z + 3 log x - 2 log y
(multiplication is addition and division is subtraction).
Trig function of sinθ/tanθ equal to?.
cosθ
(convert everything to sin and cos and simplify).
The horizontal asymptote of f(x) = (x2 -15x + 54)/(9x + 3).
DNE
(because the degree of the numerator is greater than the denominator, there is no horizontal asymptote).
f(x) = sine/x when x=1
f(x) = 3x/x when x does not equal to 1
DNE
(LHS does not equal to RHS limits)
The derivative of y = log(4x5).
y' = (1/4x^5) *(20x^4)
(derivative of log)
All possible values of x in log2x(3x2 + 8x - 15) = 2.
3 and 5
(convert to exponential form and solve the quadratic).
The value of sin 2θ given that sin θ is 12/13.
Hint: sin 2θ = 2sinθcosθ
120/169
(follow the formula)
The coordinate point of the hole in the function f(x) = (3x + 18)/(x2 +15x + 54)
(-6,1)
(calculating holes algebraically)
what is limit of x approaching to 0 of f(x) when f(x) equals to (1-cos(x))/x
0
The dy/dx given -sin(x + y) + x3.
dy/dx =3x2 - cos(x + y)(1-d/d(x)(y))
(implicit differentiation)