True/False
d/dx (ln(3x)) = 1/(3x)
False
Differentiate the function x^2 = p^2 + 55 with respect to t
2x(dx)/(dt) = 2p(dp)/(dt)
f(x) = (x^2)/(x^2 - 9) , state the vertical and horizontal asymptotes of f
Vertical asymptotes at x = -3 and x = 3, and horizontal asymptote at y = 1
find the derivative of f(x) = e^(6x^2)
f'(x) = 12xe^(6x^2)
if f'(x) < 0 on (a,b), then f is decreasing on (a,b)
True
Differentiate the function 2x + 2y = x^2 + y^2 with respect to x
(dy)/dx = (2x-2)/(2-2y)
find at which x-values the inflection point(s) of f(x) = 2x^4 - 12x^2 occur
x = 1, x = -1
g(t) = 4ln(3t+5) , find the derivative of the function
g'(t) = 12/(3t+15)
A critical point for the function f is in the domain of f
True
suppose y = 2x^3 and (dx)/dt = 2 at x = 4, what is the value of (dy)/dt
192
Find the intervals where f(x) = 2x^3 - 12x^2 +18x is increasing and decreasing
the function f is increasing on (-oo, 1)U(3, oo) and is decreasing on (1,3)
find the derivative of f(x) = e^x/x^5 + e^(3x)
f'(x) =
Using differentials, approximate the value of √9.9 to 3 decimal places. (you will get this wrong if you simply plug √9.9 into your calculator)
3.150
The function f(x) = x^3 has an absolute minimum and maximum value
False
given that y depends of x, find y' of 6xy + 5y^2 = 4 + 3x^2
y' = (6(x-y))/(10y + 6x)
Find the critical points of the function f(x) = x^2/(x-1)
x = 0, x = 2
find the derivative of f(x) = (6x^3+5x^2)(ln(x))
f'(x) = (6x^3+5x^2)(1/x)+ln(x)(18x^2+10x)
A business employs four workers, each is paid $30000 per year. The business has additional costs of $( 50x+2x^2 ) for materials where x is the number of products created. The revenue function is given by $( 50x^3+5x^2 ). Find the profit function. (Hint: P(x) = R(x) - C(x))
P(x) = 50x^3 + 3x^2-50x-120000
If f(c) exists, f'(c) = 0 and f''(c) > 0, then f(x) must have a relative maximum at x = c
False, it would be a relative minimum
The demand of boxes of candy per year is given by:
x^2 = p^2 +55
where p is the price in dollars and x is the quantity of boxes of candy demanded. Find the rate of change of the price per year when the quantity demanded is increasing at a rate of 4 boxes of candy per year at the point where the price is $3
(dp)/(dt) = 64/6 or 10.67 dollars per year
using logarithmic differentiation, find y' of y = x^(6x^3+5x^2)
y' = x^(6x^3+5x^2)((6x^3+5x^2)(1/x)+ln(x)(18x^2+10x))
The average net profit of a car salesperson for selling x cars per day (on average) is given by P(x) = -2x^3 + 15x^2 -25 , on the interval [0,6] find the number of cars the salesperson should sell per day to maximize their average profit. (hint: optimize using the first derivative)
P(x) is greatest when x = 5 (P(5) = 100)