1. Find each limit, SHOWING WORK.
a) x→-3lim (x2 - 9)/(x2 + 7x +12)
b) x→∞lim (5x - 6x2)/(4x2 - x + 9)
c) x→-4lim 2x3 - 5x2 + 7x
d)x→3lim (x2 + 11)/(x2 + 7x +10)
2. Credit will only be awarded for part (b) in the case of following the directions stated.
a) STATE the full Definition of the Derivative in formula format.
b) Use the Definition of the Derivative to determine the derivative of each function:
(i) f (x) = 5 / (3 - 2x)
(ii) f (x) = -5 / (x + 6)
(iii) f (x) = 2x2 - 7x + 4
(iv) g (x) = √(x-2)
3. Find the Slope-Intercept Equation of the line tangent to f(x) at x = -3, f (x) = -x3 - 7x2 + 4
4. Find the derivative of each function, SHOWING ALL WORK. Definition NOT needed.
a) y = 3x5 - 4x-6 + (7/√2)
b) h(x) = (2x2 - 5)/ (x2 + x)
c) h(x) = (5x2 + 8)6
5. Use the first derivative of the given function to: R(x) = (x3 - 16)/x
a) Locate all critical values
b) Determine all Intervals of Inc and Dec, showing a proper Sign Chart
c) Determine the coordinates of all Local Extrema, naming which type
6. Find derivatives of each function by use of Power Rule (Definition is NOT needed)
a) f (x) = 2x3 + (3/x4)
b) g(x) = 4√x
7. Find the Absolute Maximum and Absolute Minimum for f (x) = x3 + 5x2 - 8x + 6 on [-6, 4]
8. Use the given cost and revenue functions to perform the requested tasks.
C(x) = 60x + 20000, R(x) = -0.5x2 + 400x
a) Find the Profit function, P(x) =
b) Find the Marginal Profit function
c) Use Marginal Profit to estimate the profit generated by the 101st item produced.
9. Use Implicit Differentiation to find the derivative of:
(a) 4x3 y5 = 8x - 9y2 +12 (b) 3x2 + x3 y4 = 8y - 7
10. List ALL Critical Points of f (x) = x4 + 4x3 - 36x2
a) Using the Critical Points, build a Sign Chart and determine Intervals of Inc/Dec, showing them in Interval Notation
b) Identify each Critical Point as a Rel Max, Rel Min, or Neither in Coordinate Form
11. Find the 1st and 2nd Derivatives of h(x) and use h"(x) to determine Intervals of Concavity and then identify the coordinates of any Inflection Points. Intervals in Interval Notation.
h(x) = x5 - (40 x3/3) + 8x
12. A giant sphere-shaped ice ball is melting uniformly. If the ice is melting such that the radius is shrinking by 10cm per hour, find the rate at which volume is shrinking when the radius is 250cm.
(Use : V = (4Πr3/3) Round answers to 2 decimal places.
13. A motel currently rents out 300 rooms per night at $60 per room. The manager believes that for every $4 increase in the room rate, 10 less rooms will be rented each night. Find:
a) A function to represent the nightly revenue from room rentals, R(x), where x represents the number of $4 room price increases
b) Find the Maximum value for R(x) using Calculus techniques
c) Determine the room rental price and number of rooms rented at this maximized point
14. Graph:
At what values x (if any) is G(x) not continuous?
15. A company's Profit function is given by P(x) = -0.04x2 + 280x - 50000
i) Find the actual difference in total profit if they increase production from 2400 units to 2450
ii) Find the estimated difference in total profit if they increase production from 2400 to 2450 using the techniques in section 2.5 of the text
16. Given: x2 + 3xy + y2 = 11; dx/dt = 2 when x = 1 and y = 2. Find dy/dt
17. Find the Absolute Maximum and Absolute Minimum (if they exist) for y = x + (25/x) on (-∞, 0)
18. Find the Absolute Maximum and Absolute Minimum for f (x) = x3 + 4x2 - 3x on [-4,1]
19. Find the Absolute Maximum and Absolute Minimum for f (x) = x4 - 4x3 + 5 on [-1, 2]
20. List ALL Asymptotes of each function below
a) R(x) = (x2 -16x) / (2x2 + 6x)
b) h(x) = (x - 4) / (x2 - 9)
c) Q(x) = (x3 - 4x2) / (x2 + 3x)
21. Use each Demand function, where X = price, to determine the Elasticity at the given X
a) q = D(x) = 400 - 6x at (i) x = 50, (ii) x = 20
b) q = D(x) = 200 / (x + 40)2 at (i ) x = 60, (ii) x = 20
22. Widget World sells 900 widgets each year. Their storage costs are $9 per widget for a year's storage. Ordering costs are $50 per order and $12 for each widget. Assuming steady sales of widgets over the course of a year, build a total cost function and use it to determine what Lot Size, X, should be ordered each time for lowest total costs. Find that minimum total cost.
23. Perform all of the listed steps below for:
h (x) = 2x / (x2 + 4)
a) Find h'(x) and ALL Critical Pts. Build a Sign Chart to determine Intervals of Inc/Dec
b) Name coordinates of any Relative Max/Min's
c) Find h"(x) and any possible Inflection Pts. Build a Sign Chart to determine Intervals of Concavity
d) Name coordinates of any Inflection Pts
e) Sketch ALL Asymptotes and the use the information from (a)-(d) to sketch the graph of h(x)
24. Find the Average Rate of Change of f (x) = (1/4) x2 + 2x + 4 on each Interval below:
i) x = 3 to x = 5
ii) x = 3 to x = 3.5
iii) x = 3 to x = 3.1
iv) x = 3 to x = 3.01
Use your answers from (i) to (iv) to predict the value of f '(3), and then check against the real f '(3).