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Sample Final Examination

1. Consider the function y(x) defined implicitly by [pic]. Find the equation of the line tangent to the graph at the point (4, 0). Use your equation to find an approximate value for y when x = 4.1.

2. The graphs of function f, its antiderivative, F, and its second derivative, [pic]are shown in this plot:

| |Write labels here (f, F, or [pic]) |

|[pic] | |

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| | |

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| | |

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| | |

| | |

Identify each curve and briefly justify each of your choices. {N.B.: Do not justify your third choice by saying something like “well...that’s the only choice left.” Indicate why your choice is correct.}

3. A psychological study of memory and memorization skills tested how quickly the “average” person can memorize a list of words. They found that the number of words, W, that can be memorized in t minutes is given by:

[pic]

At t = 5 minutes, what is the rate (in words/minute) at which the average person in the study can memorize words?

4. A holiday turkey is placed in a 325°F oven to roast. The turkey is initially at room temperature, 70°F.

a) Assume that the turkey takes 5 hours to reach a temperature of 180°F. Newton’s Law of Heating gives the temperature of the turkey at any time t after it has been placed in the oven as

T(oven) – T(turkey) = [pic].

Find the constants C and k. and write the equation.

(b) Use the function found in part (a) to determine at what rate the turkey’s temperature is changing one hour after it is put into the oven.

5. A cyclist is on a long straight road leading from her home to a nearby lake. Assume that the speed of travel toward the lake is taken as positive (and speed toward home is negative) The graph below shows the cyclist’s speed as a function of time. She reaches the lake at the point labeled A, stops for lunch, and leaves for home at the point labeled B.

(a) determine the distance from the cyclist’s home to the lake; and,

(b) determine the total distance the cyclist traveled during her trip.

[pic]

6. The table below shows the values of two different functions. One is a power function and one is an exponential function.

x f(x) g(x)

1.2 5.184 0.977

1.4 8.232 1.095

1.6 12.288 1.224

(a) Find an equation for each of the functions.

(b) Find an equation for the function h(x) shown in the graph:

[pic]

7. Consider two functions f and g. The graphs of their derivatives are given below. For each given statement answer true or false (circle the appropriate word) and include a very brief explanation.

[pic] [pic]

[pic] [pic]

[pic] TRUE FALSE

[pic] TRUE FALSE

[pic] TRUE FALSE

[pic] TRUE FALSE

[pic] TRUE FALSE

[pic] TRUE FALSE

8. The regular airfare between Philadelphia and Phoenix is $500. One airline flies the route using 747’s with a capacity of 380 passengers. The airline does a study and finds that their average flight carries 300 passengers. The study also revealed that for every $20 reduction in the airfare, 20 more passengers would take each flight.

a) If x is the number of passengers, write an equation for the revenue, r(x)

(b) Use your equation from (a) and some calculus to find the fare that maximizes the revenue (solution by other means will receive no credit).

9. Evaluate the integrals:

a) [pic]

b) [pic]

10. An aspiring rocket scientist launches a model rocket from ground level at time t =0. The rocket goes straight up. The graph below shows the velocity of the rocket as a function of the time since the rocket was launched:

[pic]

• Sketch a graph of the acceleration of the rocket as a function of time

[pic]

• Sketch a graph of the height of the rocket as a function of time (h = 0 at launch

[pic]

• Let v(t) be the rocket’s velocity as a function of time. From the graph of the rocket’s velocity (above), which of these quantities is larger:[pic] or [pic]

• What do you know about the sign of [pic]? What does this mean physically?

• What happens to the rocket at [pic]? at [pic]? at [pic]?

11. Find the area of the region enclosed by the curves y = x2 – 2 and y = 2x +1. Write an appropriate integral and show your work in evaluating the integral (a numerical result produced by a calculator will receive no credit).

12. Find the volume of the solid obtained by rotating the region in the plane bounded by the curves [pic] and y = 0 around the line x =2.

13. We have arctan(3) = [pic]. Use the trapezoid rule with three subdivisions to compute an approximation for arctan(3)..

14. The graph below is the graph of g((x) (the derivative of the function g(x)). Suppose we also know that g(0) = 10.

[pic]

(a) Complete the following table:

|x |0 |1 |2 |3 |4 |5 |6 |

|g(x) |10 | | | | | | |

(b) What are the maximum and minimum values of g(x) for 0 ≤ x ≤ 6, and where (i.e., for what values of x) do they occur?

(c) Where is the graph of g(x) concave up?

(d) For what values of x does the graph of g(x) have aninflection point ?

(e) Sketch the graph of g(x).

15. Let [pic] for x > 0.

(a) For what values of x does g(x) have a (local) maximum, minimum or inflection point?

(b) Compute the range of g(x).

(c) For what values of c does the equation ln x = cx have at least one solution?

(d) For what values of c does the equation ln x = cx have more than one solution ?

(e) For that values of a > 0 does the equation ax = x have a solution?

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