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f ( x ) = 4 x + 7 ; x 1 = 2 , x 2 = 5

4

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f ( x ) = 8 x 3 ; x 1 = −1 , x 2 = 3

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f ( x ) = x 2 + 2 x + 1 ; x 1 = 3 , x 2 = 3.5

8.5

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f ( x ) = x 2 + x + 2 ; x 1 = 0.5 , x 2 = 1.5

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f ( x ) = 4 3 x 1 ; x 1 = 1 , x 2 = 3

3 4

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f ( x ) = x 7 2 x + 1 ; x 1 = −2 , x 2 = 0

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f ( x ) = x ; x 1 = 1 , x 2 = 16

0.2

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f ( x ) = x 9 ; x 1 = 10 , x 2 = 13

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f ( x ) = x 1 / 3 + 1 ; x 1 = 0 , x 2 = 8

0.25

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f ( x ) = 6 x 2 / 3 + 2 x 1 / 3 ; x 1 = 1 , x 2 = 27

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For the following functions,

  1. use [link] to find the slope of the tangent line m tan = f ( a ) , and
  2. find the equation of the tangent line to f at x = a .

f ( x ) = 3 4 x , a = 2

a. −4 b. y = 3 4 x

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f ( x ) = x 5 + 6 , a = −1

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

a. 3 b. y = 3 x 1

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f ( x ) = 1 x x 2 , a = 0

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f ( x ) = 7 x , a = 3

a. −7 9 b. y = −7 9 x + 14 3

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f ( x ) = 2 3 x 2 , a = −2

a. 12 b. y = 12 x + 14

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f ( x ) = −3 x 1 , a = 4

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f ( x ) = 2 x + 3 , a = −4

a. −2 b. y = −2 x 10

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For the following functions y = f ( x ) , find f ( a ) using [link] .

f ( x ) = 5 x + 4 , a = −1

5

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f ( x ) = −7 x + 1 , a = 3

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

13

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f ( x ) = 3 x 2 x + 2 , a = 1

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f ( x ) = x 2 , a = 6

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f ( x ) = 1 x , a = 2

1 4

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f ( x ) = 1 x 3 , a = −1

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f ( x ) = 1 x 3 , a = 1

−3

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For the following exercises, given the function y = f ( x ) ,

  1. find the slope of the secant line P Q for each point Q ( x , f ( x ) ) with x value given in the table.
  2. Use the answers from a. to estimate the value of the slope of the tangent line at P .
  3. Use the answer from b. to find the equation of the tangent line to f at point P .

[T] f ( x ) = x 2 + 3 x + 4 , P ( 1 , 8 ) (Round to 6 decimal places.)

x Slope m P Q x Slope m P Q
1.1 (i) 0.9 (vii)
1.01 (ii) 0.99 (viii)
1.001 (iii) 0.999 (ix)
1.0001 (iv) 0.9999 (x)
1.00001 (v) 0.99999 (xi)
1.000001 (vi) 0.999999 (xii)

a. (i) 5.100000 , (ii) 5.010000 , (iii) 5.001000 , (iv) 5.000100 , (v) 5.000010 , (vi) 5.000001 , (vii) 4.900000 , (viii) 4.990000 , (ix) 4.999000 , (x) 4.999900 , (xi) 4.999990 , (x) 4.999999 b. m tan = 5 c. y = 5 x + 3

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[T] f ( x ) = x + 1 x 2 1 , P ( 0 , −1 )

x Slope m P Q x Slope m P Q
0.1 (i) −0.1 (vii)
0.01 (ii) −0.01 (viii)
0.001 (iii) −0.001 (ix)
0.0001 (iv) −0.0001 (x)
0.00001 (v) −0.00001 (xi)
0.000001 (vi) −0.000001 (xii)
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[T] f ( x ) = 10 e 0.5 x , P ( 0 , 10 ) (Round to 4 decimal places.)

x Slope m P Q
−0.1 (i)
−0.01 (ii)
−0.001 (iii)
−0.0001 (iv)
−0.00001 (v)
−0.000001 (vi)

a. (i) 4.8771 , (ii) 4.9875 (iii) 4.9988 , (iv) 4.9999 , (v) 4.9999 , (vi) 4.9999 b. m tan = 5 c. y = 5 x + 10

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[T] f ( x ) = tan ( x ) , P ( π , 0 )

x Slope m P Q
3.1 (i)
3.14 (ii)
3.141 (iii)
3.1415 (iv)
3.14159 (v)
3.141592 (vi)
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[T] For the following position functions y = s ( t ) , an object is moving along a straight line, where t is in seconds and s is in meters. Find

  1. the simplified expression for the average velocity from t = 2 to t = 2 + h ;
  2. the average velocity between t = 2 and t = 2 + h , where (i) h = 0.1 , (ii) h = 0.01 , (iii) h = 0.001 , and (iv) h = 0.0001 ; and
  3. use the answer from a. to estimate the instantaneous velocity at t = 2 second.

s ( t ) = 1 3 t + 5

a. 1 3 ; b. (i) 0. 3 m/s, (ii) 0. 3 m/s, (iii) 0. 3 m/s, (iv) 0. 3 m/s; c. 0. 3 = 1 3 m/s

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s ( t ) = 2 t 3 + 3

a. 2 ( h 2 + 6 h + 12 ) ; b. (i) 25.22 m/s, (ii) 24.12 m/s, (iii) 24.01 m/s, (iv) 24 m/s; c. 24 m/s

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s ( t ) = 16 t 2 4 t

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Use the following graph to evaluate a. f ( 1 ) and b. f ( 6 ) .

This graph shows two connected line segments: one going from (1, 0) to (4, 6) and the other going from (4, 6) to (8, 8).

a. 1.25 ; b. 0.5

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Use the following graph to evaluate a. f ( −3 ) and b. f ( 1.5 ) .

This graph shows two connected line segments: one going from (−4, 3) to (1, 3) and the other going from (1, 3) to (1.5, 4).
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For the following exercises, use the limit definition of derivative to show that the derivative does not exist at x = a for each of the given functions.

f ( x ) = x 1 / 3 , x = 0

lim x 0 x 1 / 3 0 x 0 = lim x 0 1 x 2 / 3 =

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f ( x ) = x 2 / 3 , x = 0

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f ( x ) = { 1 , x < 1 x , x 1 , x = 1

lim x 1 1 1 x 1 = 0 1 = lim x 1 + x 1 x 1

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f ( x ) = | x | x , x = 0

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[T] The position in feet of a race car along a straight track after t seconds is modeled by the function s ( t ) = 8 t 2 1 16 t 3 .

  1. Find the average velocity of the vehicle over the following time intervals to four decimal places:
    1. [4, 4.1]
    2. [4, 4.01]
    3. [4, 4.001]
    4. [4, 4.0001]
  2. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at t = 4 seconds.

a. (i) 61.7244 ft/s, (ii) 61.0725 ft/s (iii) 61.0072 ft/s (iv) 61.0007 ft/s b. At 4 seconds the race car is traveling at a rate/velocity of 61 ft/s.

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[T] The distance in feet that a ball rolls down an incline is modeled by the function s ( t ) = 14 t 2 , where t is seconds after the ball begins rolling.

  1. Find the average velocity of the ball over the following time intervals:
    1. [5, 5.1]
    2. [5, 5.01]
    3. [5, 5.001]
    4. [5, 5.0001]
  2. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at t = 5 seconds.
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Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by s = f ( t ) and s = g ( t ) , where s is measured in feet and t is measured in seconds.

Two functions s = g(t) and s = f(t) are graphed. The first function s = g(t) starts at (0, 0) and arcs upward through roughly (2, 1) to (4, 4). The second function s = f(t) is a straight line passing through (0, 0) and (4, 4).
  1. Which vehicle has traveled farther at t = 2 seconds?
  2. What is the approximate velocity of each vehicle at t = 3 seconds?
  3. Which vehicle is traveling faster at t = 4 seconds?
  4. What is true about the positions of the vehicles at t = 4 seconds?

a. The vehicle represented by f ( t ) , because it has traveled 2 feet, whereas g ( t ) has traveled 1 foot. b. The velocity of f ( t ) is constant at 1 ft/s, while the velocity of g ( t ) is approximately 2 ft/s. c. The vehicle represented by g ( t ) , with a velocity of approximately 4 ft/s. d. Both have traveled 4 feet in 4 seconds.

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[T] The total cost C ( x ) , in hundreds of dollars, to produce x jars of mayonnaise is given by C ( x ) = 0.000003 x 3 + 4 x + 300 .

  1. Calculate the average cost per jar over the following intervals:
    1. [100, 100.1]
    2. [100, 100.01]
    3. [100, 100.001]
    4. [100, 100.0001]
  2. Use the answers from a. to estimate the average cost to produce 100 jars of mayonnaise.
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[T] For the function f ( x ) = x 3 2 x 2 11 x + 12 , do the following.

  1. Use a graphing calculator to graph f in an appropriate viewing window.
  2. Use the ZOOM feature on the calculator to approximate the two values of x = a for which m tan = f ( a ) = 0 .

a.
The function starts in the third quadrant, passes through the x axis at x = −3, increases to a maximum around y = 20, decreases and passes through the x axis at x = 1, continues decreasing to a minimum around y = −13, and then increases through the x axis at x = 4, after which it continues increasing.
b. a 1.361 , 2.694

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[T] For the function f ( x ) = x 1 + x 2 , do the following.

  1. Use a graphing calculator to graph f in an appropriate viewing window.
  2. Use the ZOOM feature on the calculator to approximate the values of x = a for which m tan = f ( a ) = 0 .
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Suppose that N ( x ) computes the number of gallons of gas used by a vehicle traveling x miles. Suppose the vehicle gets 30 mpg.

  1. Find a mathematical expression for N ( x ) .
  2. What is N ( 100 )? Explain the physical meaning.
  3. What is N ( 100 ) ? Explain the physical meaning.

a. N ( x ) = x 30 b. 3.3 gallons. When the vehicle travels 100 miles, it has used 3.3 gallons of gas. c. 1 30 . The rate of gas consumption in gallons per mile that the vehicle is achieving after having traveled 100 miles.

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[T] For the function f ( x ) = x 4 5 x 2 + 4 , do the following.

  1. Use a graphing calculator to graph f in an appropriate viewing window.
  2. Use the nDeriv function, which numerically finds the derivative, on a graphing calculator to estimate f ( −2 ) , f ( −0.5 ) , f ( 1.7 ) , and f ( 2.718 ) .
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[T] For the function f ( x ) = x 2 x 2 + 1 , do the following.

  1. Use a graphing calculator to graph f in an appropriate viewing window.
  2. Use the nDeriv function on a graphing calculator to find f ( −4 ) , f ( −2 ) , f ( 2 ) , and f ( 4 ) .

a.
The function starts in the second quadrant and gently decreases, touches the origin, and then it increases gently.
b. −0.028 , −0.16 , 0.16 , 0.028

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Practice Key Terms 4

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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