Review Example 3: Suppose that we know the revenues of a company each year since This information is given in the table below:

Transcription

1 Math 1314 ONLINE Final Exam Review Review Example 1: Suppose 3 g( x) = x x 9x Find the zeros of the function. Review Example : Find any points where intersect. f ( x) = 1.45x 7.x 1.6 and g( x) =.84x 1.9 Review Example 3: Suppose that we know the revenues of a company each year since 005. This information is given in the table below: year revenues ( in millions of dollars) A. Create a scatterplot and determine which of the regression models are good candidates for this data. B. Find regression models for each, and find the related values for r or R. C. Which model would be the best one to use? Why? D. Use that model to predict revenues in 01.

2 Review Example 4: Evaluate: x + 8 lim. x 4 x 4 Review Example 5: Evaluate: lim( 4x 7x + ) x 3 Review Example 6: Determine if the function given is indeterminate. If it is, use GGB to evaluate the limit. x + 3x 10 lim x x 4 Review Example 7: Evaluate: 5x + x + 6 lim x x 7x + 1 Review Example 8: Evaluate : lim x 7 x x + 9x 3 4 3x 4x + 5 Review Example 9: Evaluate: lim x x 7x

3 x + 4, x > Review Example 10: Suppose f ( x) =. x 1, x (if it exists). Find lim f ( x), lim f ( x) and lim f ( x) x + x x 5x 1, x 1 Review Example 11: Determine if f ( x) = is continuous at x = 1. x + 3, x < 1 Review Example 1: Find the derivative of f ( x) = x + 6x + 3 using the limit definition. Review Example 13: Suppose f ( x) = 3x 4x 1. Find the average rate of change of f over the interval [, 5].

4 4 3 6 Review Example 14: Find the derivative: f ( x) = 5x + 3x 4. x Review Example 15: Find the numerical derivative of 3 f ( x) = x x when x = 4. Review Example 16: Find the numerical derivative of f ( x) = x at x =. x + 1 Review Example 17: Write an equation of the line that is tangent to when x = f ( x) = x x + 7x Review Example 18: A country s gross domestic product (in millions of dollars) is modeled by 3 the function G( t) = t + 45t + 0t where 0 t 11 and t = 0 corresponds to the beginning of (A) At what rate was GDP changing at the beginning of 00? At the beginning of 004? At the beginning of 009? (B) What was the average rate of growth of the GDP over the period ?

5 Review Example 19: Suppose you are standing on the top of a building that is 8 meters high. You throw a ball up into the air, with initial velocity of 10. meters per second. Write the equation that gives the height of the ball at time t. Then use the equation to find the velocity of the ball when t =. Review Example 0: Find all values of x for which f '( x ) = 0 : 1 5 f x x x x 3 3 ( ) = Review Example 1: Find the value of the second derivative when x = : 4 3 f ( x) = 3x 5x + 7x + 1 Review Example : Suppose a company can model its costs according to the function 3 C( x) = x 0.04x + 00x + 70, 000 where C( x) is given in dollars and demand can be modeled by p = 0.0x Write the revenue function and find the smallest positive quantity for which all costs are covered.

6 Review Example 3: The quantity demanded of a certain electronic device is 8000 units when the price is $60. At a unit price of $00, demand increases to 10,000 units. The manufacturer will not market any of the device at a price of $100 or less. However for each $50 increase in price above $100, the manufacturer will market an additional 1000 units. Assume that both the supply equation and the demand equation are linear. Find the supply equation, the demand equation and the equilibrium quantity and price. Review Example 4: The weekly demand for a certain brand of DVD player is given by p =. 0x + 300, 0 x 15,000, where p gives the wholesale unit price in dollars and x denotes the quantity demanded. The weekly cost function associated with producing the DVD players is 3 given by C ( x) = x 0.04x + 00x + 70, 000. Compute C' (3000), R' (3000) and P '(3000). Interpret your results

7 Review Example 5: Suppose p = 0.04x (A) Find the elasticity of demand. (B) Find E (50) and interpret the results. (C) Find E (100) and interpret the results. (D) If the unit price is $50, will raising the price result in an increase in revenues or a decrease in revenues? (E) If the unit price is $100, will raising the price result in an increase in revenues or a decrease in revenues? Review Example 6: A biologist wants to study the growth of a certain strain of bacteria. She starts with a culture containing 5,000 bacteria. After three hours, the number of bacteria has grown to 63,000. How many bacteria will be present in the culture 6 hours after she started her study? What will be the rate of growth 6 hours after she started her study? Assume the population grows exponentially and the growth is uninhibited.

8 Review Example 7: At the beginning of a study, there are 50 grams of a substance present. After 17 days, there are 38.7 grams remaining. How much of the substance will be present after 40 days? What will be the rate of decay on day 40 of the study? Assume the substance decays exponentially. Review Example 8: Analyze the function: 3 f x x x x 4 3 ( ) = + 1 +

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11 Review Example 9: The graph given below is the first derivative of a function, f. Find the interval(s) on which the function is increasing, the interval(s) on which the function is decreasing, the x coordinate of each relative extremum (and state whether it is a relative maximum or a relative minimum), the interval(s) on which the function is concave upward, the interval(s) on which the function is concave downward and the x coordinate of any inflection points.

12 Review Example 30: An apartment complex estimates that the revenues realized from rending out x of its 100 one-bedroom apartments can be modeled by the function R( x) = 1x + 11 x. How many one bedroom apartments should be rented to maximize the revenue? What is the maximum revenue? Review Example 31: Use left endpoints and 4 subdivisions of the interval to approximate the area under f ( x) = x + 1 on the interval[0, ]. Review Example 3: Approximate the area between x axis and the graph of 3 f ( x) = 0.3x 0.807x 3.5x on [-.8, 1.33] with A. 50 rectangles and midpoints endpoints. B. 10 rectangles and right endpoints. Review Example 33: (Be able to do this by hand!!) Evaluate: 1 3 ( 3x + 4x 7) dx

13 Review Example 34: Evaluate ( ) 3 5 4x x 3 0 dx A study of worker productivity shows that the rate at which a typical Review Example 35: worker can produce widgets on an assembly line can be expressed as N( t) = 3t + 1t + 15 where t gives the number of hours after a worker s shift has begun. Determine the number of widgets a worker can produce during the first hour of his/her shift. Determine the number of widgets a worker can produce during the last hour of a five hour shift. Review Example 36: The median price of a house in a city in Arizona can be approximated by 3 the function f ( t) = t 7t + 17t where the median price is given in thousands of dollars and t is given as the number of years since 000. This function has been shown to be valid for the years 000 to 005. Determine the average median price of a home in this city during this time period.

14 Review Example 37: Find the area between the functions the interval 1 x 3. f x ( ) 3x = + and g( x) = x 3 on Review Example 38: Find the area of the region that is completely enclosed by the graphs of the functions f ( x) = x 3x and g( x) = 1.6x. Review Example 39: Suppose f ( x, y) = 3x y 4xy + 6. Compute f ( 0,0), f (, 1) and f ( 1, 3). Review Example 40: Find the domain: f ( x, y) x + 5y = x y

15 Review Example 41: Find the first partial derivatives of the function 3 f ( x, y) = 5x y x y + 9x 14y Review Example 4: Find the second-order partial derivatives of the function f ( x, y) = 3x x y + 5xy + 6y. Review Example 43: Evaluate the first and second-order partial derivatives of f ( x, y) = 3x x y + 5xy + 6y at the point (1, ).

16 Review Example 44: Find the relative extrema of the function f ( x, y) = x + xy + y 4x + 8y 1. Review Example 45: Find the relative extrema of the function (, ) 3 f x y = x xy + y + 5.