Precalculus Jeopar(dy/dx)

Logarithms

100

Can you compute log of zero or negative num?

NO

200

The x value in 3^3x = 27^-3.

 -3

(rewrite with a common base of 3 and distributing the powers to solve for x).

300

The value of x in log₂(8x) - 3log₂(6) = 1.

 27

(use the power rule and then evaluate the exponent).

400

The expanded form of log (z²x³)/y².

 2 log z + 3 log x - 2 log y

(multiplication is addition and division is subtraction).

500

All possible values of x in log_2x(3x² + 8x - 15) = 2.

 3 and 5

(convert to exponential form and solve the quadratic).

Trigonometry

100

The angle 3π/4 radians in degrees.

 135 degree

(replace pi by 180 and multiply).

200

The exact value of sin π/6.

1/2

(convert to degrees and solve using SOHCAHTOA or prior knowledge).

300

The value of sec A given that csc A = 13/12.

 13/5

(use the pythagorean theorem and plug in the values).

400

Trig function of sinθ/tanθ equal to?.

 cosθ

(convert everything to sin and cos and simplify).

500

The value of sin 2θ given that sin θ is 12/13.

Hint: sin 2θ = 2sinθcosθ 

120/169

(follow the formula)

Holes and Asymptotes

100

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).

200

The vertical asymptote of y = (x + 2)/(2x - 10).

x = 5

(vertical asymptotes are found by setting the denominator equal to 0 after simplifying).

300

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).

400

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).

500

The coordinate point of the hole in the function f(x) = (3x + 18)/(x² +15x + 54)

 (-6,1)

(calculating holes algebraically)

Limits and Continuity

100

The limit as x approaches 2 of (3x + 2x²).

 14

(basic limit computation)

200

The limit as x approaches infinity of 2x/x³.

 0

(limit transformations)

300

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)

400

find the limit of f(x) when x is approaching to 1 where 

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)

500

what is limit of x approaching to 0 of f(x) when f(x) equals to (1-cos(x))/x

 0

Differentiation

100

The derivative of y = x³.

y' = 3x²

(basic power rule)

200

The derivative of y = -sqrt(x) - [sqrt(x³)]/3.

 y' = -1/2sqrt(x) - sqrt(x)/2

(chain rule but a little more complicated)

300

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)

400

The derivative of y = log(4x⁵).

 y' = (1/4x^5) *(20x^4)

(derivative of log)

500

The dy/dx given -sin(x + y) + x³.

dy/dx =3x² - cos(x + y)(1-d/d(x)(y))

(implicit differentiation)