1.    (6 points) Let f(x) = 3 − 2x + 3x2. Use the limit definition of the derivative to find f ′ (x). Show all steps, especially the algebra involved, and use proper notation.

2.    (3 points each) For all parts of this question, circle the correct answer. (No partial credit will be given for incorrect answers).

(a) The derivative off(x) = ∫ (2 (3)x 2 )dx is:

(i)   x3

2

(ii)  2 (3)x

(iii) 2 (3)x 2 + C

(iv)   (4x)

(v)  none of the other options.

(b) The derivative of the function f(x) = 5(√5x 7  )(3)x  is

(i)   5( √5x 7 )(3x )ln3 + 7(√5x 2 )(3x )

(ii)  7( √5x 2 )(ln3)3x

(iii) 5(√7x 5 )(3x )ln3 + 7( 2√x 5 )(3x)

(iv) 35(√5x 7 )(3x)(lnx)( 5√x 2 )(3x)

(v)  none of tℎe otℎer options

(c) The derivative of the function g(x) =  is

(i)   − 3(2x  − 2 ln(x))−2 (2x  ln(2) − 2x−1)

(ii)  − 3(2x ln(2) − 2x−1)−2

(iii) 3 ln(2x  − 2 ln(x)) (2x  ln(2) − 2x−1)

(iv) 3 ln(2x  ln(2) − 2x−1)

(v)  none of tℎe otℎer options

(d) The general antiderivative of the function ℎ(t) =  + √t2  + 1) is

(i)   −t−1  + √t2  + 1 + C

(ii)   t −3  + √t 2  + 1 + C

(iii) − 2t−3  − t + ln |t| + C

(iv)  t 2  +  

(v)  none of tℎe above

(e) The general antiderivative of f(x) = − 8x−3  + 0.29(0.3)x  + 5 is:

(i)   x4(24) + 0.29(ln0.3)(0.3)x  + 5x + C

(ii)   + 0.29(ln0.3)(0.3)x  + C

(iii)  +  +5x + C

(iv) x4(24) + 0.29(ln0.3)(0.3)x  + C

(v)  none of the other options

3.    (4 points each) Last year, the price (in dollars) of an "epic sandwich” at Uncle Grover’s Snack Bar varied according to the function p(x) = − 0.01x3  + 1.1x + 6, where x denotes time in months  since the beginning of the year. Round off your answers to 2 decimal places.

a.    Use the Fundamental Theorem of Calculus to evaluate   p(x)dx.  Be sure to include units in your answer.

b.   Write a sentence interpreting your answer to Part a in the context of the problem.

4.    (6 points) Suppose f is a function whose derivative is given by f ′ (x) = 2x3  + 8x2  − 42x .  Find the x-coordinate(s) of the critical point(s) of the function f(x).  Use the First Derivative Test to   determine which of the critical points are relative maximum points (if any) and which are relative minimum points (if any). Show work.

5.    (6 points) Calculate the derivative of the function

f(x) = 123x2−8x  −  .

Show all the steps and use proper notation.

6.    (10 points) A group of students has determined that the demand for a pair of pants sold for x dollars can be modeled by the function D(x) = 10,000 − 10x2  when the price is kept between

$0 and $30. Include units in all your answers to the following questions.

(a)  Write the formula for revenue.

(b)  Find the revenue when the selling price is $12 per pair of pants.

(c)  Write the formula for the rate of change of revenue.

(d)  Find the rate of change of revenue when the selling price is $12 per pair of pants.

(e)  Using your answers from parts (b) and (d), estimate the revenue if the selling price is increased to $12.50 per pair of pants.

7.    (8 points) Suppose the rate of change of the price of the tech- stock is given by p(t) =   4 3√t + 5 dollars per year, where  t  is the number of years from 2010. Find the model  P(t)for the price of a stock if in 2018 its price was 110 dollars. Give units.

8.(10 points) The US consumer report found that the house heating prices go on a roller-coaster ride from 2010 to 2020. The residential heating price in dollars per gallon, for selected years since 2010 are reflected in the table below:

(a) Let P(x) denote the residential heating price in  $  gallon ,  x years since 2010. Use the table above to find the best fit model for  P(x) .  Give your model for  P(x) with all coefficients  rounded to three decimal places, and with units.

(b) Using the table find the average rate of change of the price from 2010 to 2014. Write your answer to two decimal places and give units.

(c) Using the full model for P(x) , how rapidly is the residential heating price changing in year 2017? Write your answer to two decimal places and give units.

9. (11 points) Consider the function f(x) = 2x3 一 3x2 一 12x + 1.

a)    Find all inflection points off(x). Give both x and y coordinates. Round the coordinates to 2 decimal places

b)   Graph the function f(x) using the given window. Label the x and y coordinates of the point you found in part a).

xmin =  一3       xmax = 4      ymin =  一60              ymax = 40

c)      Indicate the nature of the inflection point. Circle the correct answer.

(i) point of fastest increase                         (iii) point of fastest decrease

(ii)  point of slowest increase                      (iv) point of slowest decrease

10. (12 points) The Demand for t-shirts can be modelled by D(x), where x is the selling price in dollars per t-shirt and the maximum price of a t-shirt is $35.

D(x) =

Check that D(5)  =  3965.7456642325

a)    Give a model, for the revenue R(x).

b)   Sketch the graph R′ (x), 0 ≤ x  ≤ 35 and indicate the critical point of  R(x) on this graph by “CP”

c)    Using appropriate calculator procedures, find the selling price, x, that maximizes revenue. Round your answer to four decimal places. Give units.

Round the answer from part c to 2 decimal places and find the revenue at that price. Round your answer to 2 decimal places                                        

d)    Use the first derivative test to show that the price you found in part c) indeed yields a maximum revenue.