MATH 115 Midterm 2 Review

True/False

100

d/dx (ln(3x)) = 1/(3x)

False

200

if f'(x) < 0 on (a,b), then f is decreasing on (a,b)

True

300

A critical point for the function f is in the domain of f

True

400

The function f(x) =  x^3  has an absolute minimum and maximum value

False

500

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

3.6 (implicit differentiation and related rates)

100

Differentiate the function  x^2 = p^2 + 55  with respect to t

2x(dx)/(dt) = 2p(dp)/(dt)

200

Differentiate the function  2x + 2y = x^2 + y^2  with respect to x

(dy)/dx = (2x-2)/(2-2y)

300

suppose  y = 2x^3 and (dx)/dt = 2  at x = 4, what is the value of  (dy)/dt 

192

400

given that y depends of x, find y' of  6xy + 5y^2 = 4 + 3x^2 

y' = (6(x-y))/(10y + 6x)

500

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

4.1-4.3 (curve sketching)

100

 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

200

find at which x-values the inflection point(s) of  f(x) = 2x^4 - 12x^2  occur

x = 1, x = -1

300

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) 

400

Find the critical points of the function  f(x) = x^2/(x-1)  

x = 0, x = 2

5.4-5.5 (differentiation of log and exp. functions)

100

find the derivative of f(x) = e^(6x^2)

f'(x) =  12xe^(6x^2) 

200

 g(t) = 4ln(3t+5) , find the derivative of the function

 g'(t) = 12/(3t+15) 

300

find the derivative of  f(x) = e^x/x^5 + e^(3x) 

f'(x) = 

400

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) 

500

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)) 

Misc.

300

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

400

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 

500

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)