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12.7.1: Review Exercises

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    116465
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    Review Exercises

    Finding Limits: A Numerical and Graphical Approach

    For the following exercises, use Figure 1.

    Graph of a piecewise function with two segments. The first segment goes from (-1, 2), a closed point, to (3, -6), a closed point, and the second segment goes from (3, 5), an open point, to (7, 9), a closed point.
    Figure 1
    1.

    lim x −1 + f(x) lim x −1 + f(x)

    2.

    lim x −1 f(x) lim x −1 f(x)

    3.

    lim x1 f(x) lim x1 f(x)

    4.

    lim x3 f(x) lim x3 f(x)

    5.

    At what values of x x is the function discontinuous? What condition of continuity is violated?

    6.

    Using Table 1, estimate lim x0 f(x). lim x0 f(x).

    xx F(x) F(x)
    −0.1 2.875
    −0.01 2.92
    −0.001 2.998
    0 Undefined
    0.001 2.9987
    0.01 2.865
    0.1 2.78145
    0.15 2.678
    Table 1

    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 limit as x x approaches a, a, state it. If not, discuss why there is no limit.

    7.

    f(x)={ | x |1, if x1 x 3 , if x=1 a=1 f(x)={ | x |1, if x1 x 3 , if x=1 a=1

    8.

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

    9.

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

    Finding Limits: Properties of Limits

    For the following exercises, find the limits if lim xc f( x )=−3 lim xc f( x )=−3 and lim xc g( x )=5. lim xc g( x )=5.

    10.

    lim xc ( f(x)+g(x) ) lim xc ( f(x)+g(x) )

    11.

    lim xc f(x) g(x) lim xc f(x) g(x)

    12.

    lim xc ( f(x)g(x) ) lim xc ( f(x)g(x) )

    13.

    lim x 0 + f(x),f(x)={ 3 x 2 +2x+1 5x+3 x>0 x<0 lim x 0 + f(x),f(x)={ 3 x 2 +2x+1 5x+3 x>0 x<0

    14.

    lim x 0 f(x),f(x)={ 3 x 2 +2x+1 5x+3 x>0 x<0 lim x 0 f(x),f(x)={ 3 x 2 +2x+1 5x+3 x>0 x<0

    15.

    lim x 3 + ( 3x[x] ) lim x 3 + ( 3x[x] )

    For the following exercises, evaluate the limits using algebraic techniques.

    16.

    lim h0 ( ( h+6 ) 2 36 h ) lim h0 ( ( h+6 ) 2 36 h )

    17.

    lim x25 ( x 2 625 x 5 ) lim x25 ( x 2 625 x 5 )

    18.

    lim x1 ( x 2 9x x ) lim x1 ( x 2 9x x )

    19.

    lim x4 7 12x+1 x4 lim x4 7 12x+1 x4

    20.

    lim x3 ( 1 3 + 1 x 3+x ) lim x3 ( 1 3 + 1 x 3+x )

    Continuity

    For the following exercises, use numerical evidence to determine whether the limit exists at x=a. x=a. If not, describe the behavior of the graph of the function at x=a. x=a.

    21.

    f(x)= 2 x4 ;a=4 f(x)= 2 x4 ;a=4

    22.

    f(x)= 2 ( x4 ) 2 ;a=4 f(x)= 2 ( x4 ) 2 ;a=4

    23.

    f(x)= x x 2 x6 ;a=3 f(x)= x x 2 x6 ;a=3

    24.

    f(x)= 6 x 2 +23x+20 4 x 2 25 ;a= 5 2 f(x)= 6 x 2 +23x+20 4 x 2 25 ;a= 5 2

    25.

    f(x)= x 3 9x ;a=9 f(x)= x 3 9x ;a=9

    For the following exercises, determine where the given function f(x) f(x) is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

    26.

    f(x)= x 2 2x15 f(x)= x 2 2x15

    27.

    f(x)= x 2 2x15 x5 f(x)= x 2 2x15 x5

    28.

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

    29.

    f(x)= x 3 125 2 x 2 12x+10 f(x)= x 3 125 2 x 2 12x+10

    30.

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

    31.

    f(x)= x+2 x 2 3x10 f(x)= x+2 x 2 3x10

    32.

    f(x)= x+2 x 3 +8 f(x)= x+2 x 3 +8

    Derivatives

    For the following exercises, find the average rate of change f(x+h)f(x) h . f(x+h)f(x) h .

    33.

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

    34.

    f(x)=5 f(x)=5

    35.

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

    36.

    f(x)=ln(x) f(x)=ln(x)

    37.

    f(x)= e 2x f(x)= e 2x

    For the following exercises, find the derivative of the function.

    38.

    f(x)=4x6 f(x)=4x6

    39.

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

    40.

    Find the equation of the tangent line to the graph of f( x ) f( x ) at the indicated x x value.

    f(x)= x 3 +4x f(x)= x 3 +4x ; x=2. x=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.

    41.

    f(x)= x | x | f(x)= x | x |

    42.

    Given that the volume of a right circular cone is V= 1 3 π r 2 h V= 1 3 π r 2 h and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of π π


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