MATH 154 Practice Midterm I  Key

Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.

 

Problem 1  (15 Points)  Solve the inequality.  Write the solution on a number line.

        2x2 - 5x  >  3

Solution

We first bring all terms to the left hand side and factor

         2x2 - 5x - 3  >  0

           (2x + 1)(x - 3)  >  0

The key points occur when the left hand side equals 0 that is

        x   =   -1/2    or    x  =  3

Cut the number line into the regions 

  1. to the left of -1/2

  2. between -1/2 and 3

  3. to the right of 3

Now set up the table

Test Value

2x + 1

x-3

Total

-1

-

-

+

0

+

-

-

4

+

+

+

Since the inequality is ">" we want the positive regions.  We include the endpoints -1/2 and 3.  Our solution is 

        (-, -1/2]  U  [3, )

 

 

 

 

 

 

 

 

 

 

 

Problem 2

A. (13 Points)  Solve  x1/2 + 2x1/4 - 8  =  0  

Solution

Let    

        u  =  x1/4         u2  =  x1/2  

Substituting gives

        u2 + 2u - 8  =  0

        (u - 2)(u + 4)  =  0

        u  =  2    or    u  =  -4

Resubstituting gives

        2  =   x1/4     or    -4  =  x1/4 

        (2)4  =  (x1/4)4    or    (-4)4  =  (x1/4)4 

        x  =  16    or    x  =  256

Now we verify that 16 is a solution, however plugging 256 back in doe not yield 0.  Hence the only solution is x  =  16.

 

 

 

 

 

 

 

 

 

 

 

 

B. (13 Points)  Solve

  Solution

First isolate the square root sign

       

Now square both sides

        x - 3  =  (3 - x)2  =  x2 - 6x + 9

        x2 - 7x + 12  =  0        bringing to the right and combining like terms

        (x - 3)(x - 4)  =  0        factoring

        x  =  3        or        x  =  4

Notice that plugging in 3 works while plugging in 4 does not work.  The solution is x  =  3.

 

 

 

 

 

 

 

 

 

 

 

Problem 3  (12 Points)  Find the domain of

            x2 - 5x - 1
                                                    
          x3 - 3x2 + 2x

  Solution

        The domain of a rational function is the set of all x such that the denominator is nonzero.

             x3 - 3x2 + 2x   =  0

            x(x2 - 3x + 2)  =  0

            x(x - 2)(x - 1)  =  0

            x  =  0,    x  =  2,    or   x  =  1

so the domain is 

        {x| x  0, x  2, x 1}

 

 

 

 

 

 

 

 

 

 

 

Problem 4

Answer the following True or False.  If True, explain your reasoning, if False, explain your reasoning or show a counter-example.

A.     (7 Points)  All parabolas y = ax2 + bx + c are graphs of functions.

Solution

        True, since these parabolas pass the vertical line test.

 

 

 

 

 

 

 

 

 

 

 

 

B.     (7 Points)  If the vertex of the parabola y = ax2 + bx + c   has positive y-coordinate and the parabola is concave up, then the parabola has two x-intercepts.

Solution

False,  it will have no x-intercepts.  For example, y  =  x2 + 1 has not x intercepts.

 

 

 

 

 

 

 

 

 

 

 

 

C.     (7 Points)  If a graph has two y-intercepts then the graph is not the graph of a function.

  Solution

True, is it has two y-intercepts then there are two y values that come from the same x value.

 

 

 

 

 

 

 

 

 

 

 

 

Problem 5  Let f(x) = 3x + 2, g(x) = x + 3, and c(x) = -1.  Find

 

A) (5 Points)     f g (x)  

Solution

        f(g(x))  =  f(x + 3)  =  3(x+ 3) + 2  =  3x + 11

 

 

 

 

 

 

 

 

 

 

 

 

B) (5 Points)        f(x + h) - f(x)
                                                                  
                                    h

Solution

        f(x + h)  =  3(x + h) + 2

so

        f(x + h) - f(x)  =  3(x + h) + 2 - (3x + 2)  =  3x + 3h + 2 - 3x - 2  =  3h

Dividing by h gives

         f(x + h) - f(x)              3h
                                    =              =  3
                  h                         h

 

 

 

 

 

 

 

 

 

 

 

 

C) (5 Points)      g(f(1))     

Solution

        f(1)  =  3(1) + 2  =  5

        g(f(1))  =  g(5)  =  5 + 3  =  8

 

 

 

 

 

 

 

 

 

 

 

 

D) (5 Points)    c f (2)     

Solution

        f(2)  =  3(2) + 2  =  8

        c(8)  =  -1    c is always -1

 

 

 

 

 

 

 

 

 

 

 

E) (5 Points)           c(x)g(x)
                                                                
                            
f(x) - 7c(x)

Solution

  Plugging in, we get

             (-1)(x + 3)                -x - 3            -(x + 3)                 1
                                       =                   =                        =  -             
            3x + 2 - 7(-1)            3x + 9            3(x + 3)                3

 

 

 

 

 

 

 

 

 

 

 

 

Problem 6  (15 Points)   You are constructing a rectangular room such that one side of the room is 14 feet longer than the other side, and the distance from opposite corners is 26 feet.  What are the dimensions of the room?  Give your answer accurate to two decimal places.

  Solution

We use the Pythagorean Theorem

        x2 + (x + 14)2  =  262

        x2 + x2 + 28x + 196  =  676        Multiplying out

        2x2 + 28x - 480  =  0        combing like terms and bringing to the left

        x2 + 14x - 240  =  0        Dividing by 2

        (x + 24)(x - 10)  =  0        factoring

        x  =  10        x cannot be negative

The dimensions are 10 x 24.

 

 

 

 

 

 

 

 

 

 

 

 

 

Problem 7  (16 Points)  Graph the quadratic function.  Label any intercepts, the vertex, and the axis of symmetry.

 

        y  =  -2x2 + 4x + 6

Solution

The x-coordinate of the vertex is 

         x  =  -b/2a  =  -4/-4  =  1

Now plug 1 into the equation to get

        y  =  -2(1)2 + 4(1) + 6  =  8

Hence the vertex is at (1,8).

To find the y-intercept plug in 0 for x to get

        (0,6)

To find the x-intercept, plug in 0 for y to get

        0  =  -2x2 + 4x + 6

        0  =  x2 - 2x - 3        Dividing by -2

        0  =  (x - 3)(x + 1)    Factoring

        x  =  3  or   x  =  -1    The zero product rule

The x-intercepts are 

        (3,0)    and    (-1,0)

Notice that the coefficient of x2 is negative, so the graph is concave down. Now plot the points and sketch the graph.

 

 

 

 

 

 

 

 

 

 

 

 

 

Problem 8  The graphs of y = f(x) and y = g(x) are given below.  Find

A.     (5 Points)  f(0)

B.     (5 Points)  g(-1)

C.     (5 Points)    g f (1)  

D.     (5 Points)      f(1)
                                              
                    
g(-1)

 

  Solutions

A.  f(0) is the y-intercept of the graph of f(x) which is -1.

B.  g(-1)  =  1 since the graph goes through (-1,1).

C.  Since f(1)  =  -1, we plug -1 into gg(-1)  =  1.

D.  f(1)  =  -1, g(-1)  =  1.  Now divide to get -1/1  =  -1.

Extra Credit:  Write down one thing that your instructor can do to make the class better and one thing that you feel that the instructor should continue doing.

(Any constructive remarks will be worth full credit.)