Name  _______________________

 

AP Calculus

Summer Assignment

 

NO CALCULATOR SECTION

 

Evaluate the piecewise function at the indicated values.  (1 pt each)

 

1.                           x2 if x < 0

           f(x)   =       

x + 1 if x

 

f(-2) = ____             f(-1) = _____   f(0) = _____   f(1) = _____     f(2) = _____

 

 

 

2.                          x2 + x if x < -1

         

          f(x) =           x if

 

                             6 if x > 1

 

f(-2) = ____             f(-1) = _____   f(0) = _____   f(1) = _____     f(2) = _____

 

 

3.  Use the function f(x) = x2 – 3x + 1 to evaluate the indicated expression and simplify.  (2 pts each)

 

a.  f(x + 2)                                          b.  f(x) + f(2)

 

 

 

 

 

 

 

4.  Find f(a), f(a + h) and  where  for each below.

 (3 pts each)

 

a.  f(x) = x2 + 2                                   b.  f(x) =

 

 

 

 

 

 

 

 

5.  If f(x) = x – 2 and g(x) = x2 – 4 find each below.  (1 pt each)

 

a.  f(g(x))                                                               b.  g(f(x))

 

 

 

 

 

 

c.  f(g(0))                                                              d.  g(g(0))

 

 

 

 

 

 

 

e.  f(f(-1))

 

 

 

 

 

 

 

6.  Graph each piecewise function.  (3 pts each)

 

a.  f(x) = 2|x + 4| + 3                                             b.                1 if x < 0

                                                                                 f(x) =

                                                                                               if

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

c.                          x2 if x < 0

f(x) =                    x3 if

                   2x – 1 if x > 0

 

 

 

 

 

 

 

 

 

 

 

 

 

For 7.- 11. Write the equation of each line described.  (2 pts each)

 

  1. the vertical line through (-2, -5)

 

 

  1. horizontal line through (6, -1)

 

 

 

  1. line with undefined slope through (3, 4)

 

 

 

  1. POINT SLOPE FORM of the line through (1, -2) with slope 3.

 

 

 

  1. SLOPE INTERCEPT FORM of the line through (2, -6) with slope – ½ .

 

 

 

 

 

12.  Find the slope and y- intercept from 3x + 2y = -10.  (2 pts)

 

 

 

 

 

13.  Find the value of x for which the line connecting A(-8, -2) and B(x, 2) has a slope of 2.  (2 pts)

 

 

 

 

 

14.  Use θ = tan-1() to find the exact values for each of the SIX trig functions.  (6 pts)

 

 

 

 

 

15.  Evaluate each below.  (1 pt each)

 

a.  sin-1()                                     b.  tan(sin-1())

 

 

 

 

 

 

 

16.  Evaluate each expression.  (1 pt each)

 

a.  log                                       b.  log112 - log7

 

 

 

 

 

 

c.  log9 + log16                               d.  ln 6 – ln 15 + ln 20

 

 

 

 

 

 

 

 

 

17.  Solve each equation.  ( 3 pts each)

 

a,  x22x – 2x = 0                                            b.  e2x – 3ex + 2 = 0

 

 

 

 

 

 

 

 

c.  ln (2 + x) = 1                                            d.  log (x – 4) = 3

 

 

 

 

 

 

 

 

 

 

 

e.  log(2 – x) = 3                                         f.  log(x + 1) - log(x - 1) = 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CALCULATOR SECTION.

 

1.  A mall lake is stocked with a certain species of fish.  The fish population is modeled by P(t) =  where P(t) is the number of fish in thousands and t is measured in years since the lake was stocked.

 

    1. Find the fish population after 3 years.

(2 pts)

 

 

    1. After how many years will the fish population reach 5000 fish?

(3 pts)

 

 

 

2.  Dayton Power and Light Inc. has a power plant on the Miami River where the river is 800 feet wide.  To lay a new cable from the plant to a location in the city 2 miles downstream on the opposite side costs $180 per foot across the river and $100 per foot along the land.  (see diagram below)

 

  1. Suppose that the cable goes from the plant to a point Q on the opposite side that is x feet from the point P directly opposite the plant.  Write a function C(x) that gives the cost of laying the cable in terms of the distance x.  (3 pts)

 

 

 

  1. Generate a table of values to determine if the least expensive location from point Q is less than 2000 ft or greater than 2000 ft from P.  (4 pts)

x

C(x)

 

 

 

 

2000