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12.7.2: Practice Test

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    116466
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    Practice Test

    For the following exercises, use the graph of f f in Figure 1.

    Graph of a piecewise function with two segments. The first segment goes from negative infinity to (-1, 0), an open point, and the second segment goes from (-1, 3), an open point, to positive infinity.
    Figure 1
    1.

    f(1) f(1)

    2.

    lim x −1 + f(x) lim x −1 + f(x)

    3.

    lim x −1 f(x) lim x −1 f(x)

    4.

    lim x−1 f(x) lim x−1 f(x)

    5.

    lim x−2 f(x) lim x−2 f(x)

    6.

    At what values of x x is f f discontinuous? What property of continuity is violated?

    For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as x x approaches a. a. If the function has a limit as x x approaches a, a, state it. If not, discuss why there is no limit

    7.

    f(x)={ 1 x 3, if x2 x 3 +1,if x>2 a=2 f(x)={ 1 x 3, if x2 x 3 +1,if x>2 a=2

    8.

    f(x)={ x 3 +1, if x<1 3 x 2 1, if x=1 x+3 +4, if x>1 a=1 f(x)={ x 3 +1, if x<1 3 x 2 1, if x=1 x+3 +4, if x>1 a=1

    For the following exercises, evaluate each limit using algebraic techniques.

    9.

    lim x−5 ( 1 5 + 1 x 10+2x ) lim x−5 ( 1 5 + 1 x 10+2x )

    10.

    lim h0 ( h 2 +25 5 h 2 ) lim h0 ( h 2 +25 5 h 2 )

    11.

    lim h0 ( 1 h 1 h 2 +h ) lim h0 ( 1 h 1 h 2 +h )

    For the following exercises, determine whether or not the given function f f is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

    12.

    f(x)= x 2 4 f(x)= x 2 4

    13.

    f(x)= x 3 4 x 2 9x+36 x 3 3 x 2 +2x6 f(x)= x 3 4 x 2 9x+36 x 3 3 x 2 +2x6

    For the following exercises, use the definition of a derivative to find the derivative of the given function at x=a. x=a.

    14.

    f(x)= 3 5+2x f(x)= 3 5+2x

    15.

    f(x)= 3 x f(x)= 3 x

    16.

    f(x)=2 x 2 +9x f(x)=2 x 2 +9x

    17.

    For the graph in Figure 2, determine where the function is continuous/discontinuous and differentiable/not differentiable.

    Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-2, -1), an open point; the second segment goes from (-2, -4), an open point, to (0, 0), a closed point; the final segment goes from (0, 1), an open point, to positive infinity.
    Figure 2

    For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

    18.

    f(x)=| x2 || x+2 | f(x)=| x2 || x+2 |

    19.

    f(x)= 2 1+ e 2 x f(x)= 2 1+ e 2 x

    For the following exercises, explain the notation in words when the height of a projectile in feet, s, s, is a function of time t t in seconds after launch and is given by the function s(t). s(t).

    20.

    s(0) s(0)

    21.

    s(2) s(2)

    22.

    s'(2) s'(2)

    23.

    s(2)s(1) 21 s(2)s(1) 21

    24.

    s(t)=0 s(t)=0

    For the following exercises, use technology to evaluate the limit.

    25.

    lim x0 sin(x) 3x lim x0 sin(x) 3x

    26.

    lim x0 tan 2 (x) 2x lim x0 tan 2 (x) 2x

    27.

    lim x0 sin(x)(1cos(x)) 2 x 2 lim x0 sin(x)(1cos(x)) 2 x 2

    28.

    Evaluate the limit by hand.

    lim x1 f(x), where f(x)={ 4x7 x1 x 2 4 x=1 lim x1 f(x), where f(x)={ 4x7 x1 x 2 4 x=1

    At what value(s) of x x is the function below discontinuous?

    f(x)={ 4x7x1 x 2 4x=1 f(x)={ 4x7x1 x 2 4x=1

    For the following exercises, consider the function whose graph appears in Figure 3.

    Graph of a positive parabola.
    Figure 3
    29.

    Find the average rate of change of the function from x=1 to x=3. x=1 to x=3.

    30.

    Find all values of x x at which f'(x)=0. f'(x)=0.

    31.

    Find all values of x x at which f'(x) f'(x) does not exist.

    32.

    Find an equation of the tangent line to the graph of f f the indicated point: f(x)=3 x 2 2x6, x=2 f(x)=3 x 2 2x6, x=2

    For the following exercises, use the function f(x)=x ( 1x ) 2 5 f(x)=x ( 1x ) 2 5 .

    33.

    Graph the function f(x)=x ( 1x ) 2 5 f(x)=x ( 1x ) 2 5 by entering f(x)=x ( ( 1x ) 2 ) 1 5 f(x)=x ( ( 1x ) 2 ) 1 5 and then by entering f(x)=x ( ( 1x ) 1 5 ) 2 f(x)=x ( ( 1x ) 1 5 ) 2 .

    34.

    Explore the behavior of the graph of f(x) f(x) around x=1 x=1 by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at x=1. x=1.

    For the following exercises, find the derivative of each of the functions using the definition: lim h0 f(x+h)f(x) h lim h0 f(x+h)f(x) h

    35.

    f(x)=2x8 f(x)=2x8

    36.

    f(x)=4 x 2 7 f(x)=4 x 2 7

    37.

    f(x)=x 1 2 x 2 f(x)=x 1 2 x 2

    38.

    f(x)= 1 x+2 f(x)= 1 x+2

    39.

    f(x)= 3 x1 f(x)= 3 x1

    40.

    f(x)= x 3 +1 f(x)= x 3 +1

    41.

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

    42.

    f(x)= x1 f(x)= x1


    12.7.2: Practice Test is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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