Introduction to Derivatives: Get Stronger Answer Key

Defining the Derivative

    1. 44
    2. 8.5
    3. 34
    1. 0.2
    2. 0.25
    1. 4
    2. y=34x
    1. 3
    2. y=3x1
    1. 79
    2. y=79x+143
    1. 12
    2. y=12x+14
    1. 2
    2. y=2x10
  1. 5
  2. 13
  3. 14
  4. 14
  5. 3
    1. (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, (xii) 4.999999
    2. mtan=5
    3. y=5x+3
    1. (i) 4.8771, (ii) 4.9875, (iii) 4.9988, (iv) 4.9999, (v) 4.9999, (vi) 4.9999
    2. mtan=5
    3. y=5x+10
    1. 13
    2. (i) 0.ˉ3 m/s, (ii) 0.ˉ3 m/s, (iii) 0.ˉ3 m/s, (iv) 0.ˉ3 m/s;
    3. 0.ˉ3=13 m/s
    1. 2(h2+6h+12)
    2. (i) 25.22 m/s, (ii) 24.12 m/s, (iii) 24.01 m/s, (iv) 24 m/s
    3. 24 m/s
  6. limx0x1/30x0=limx01x2/3=
  7. limx111x1=01=limx1+x1x1
    1. (i) 61.7244 ft/s, (ii) 61.0725 ft/s, (iii) 61.0072 ft/s, (iv) 61.0007 ft/s
    2. At 4 seconds the race car is traveling at a rate/velocity of 61 ft/s.
    1. The vehicle represented by f(t), because it has traveled 2 feet, whereas g(t) has traveled 1 foot.
    2. The velocity of f(t) isconstantat[latex]1 ft/s, while the velocity of g(t) is approximately 2 ft/s.
    3. The vehicle represented by g(t), with a velocity of approximately 4 ft/s.
    4. Both have traveled 4 feet in 4 seconds.
    1. 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.
    2. a1.361,2.694
    1. N(x)=x30
    2. 3.3 gallons. When the vehicle travels 100 miles, it has used 3.3 gallons of gas.
    3. 130. The rate of gas consumption in gallons per mile that the vehicle is achieving after having traveled 100 miles.
    1. The function starts in the second quadrant and gently decreases, touches the origin, and then it increases gently.
    2. 0.028,0.16,0.16,0.028

The Derivative as a Function

  1. 3
  2. 8x
  3. 12x/li>
  4. 9x2
  5. 12x3/2
  6. The function starts in the third quadrant and increases to touch the origin, then decreases to a minimum at (2, −16), before increasing through the x axis at x = 3, after which it continues increasing.
  7. The function starts at (−3, 0), increases to a maximum at (−1.5, 1), decreases through the origin and to a minimum at (1.5, −1), and then increases to the x axis at x = 3.
  8. f(x)=3x2+2,a=2
  9. f(x)=x4,a=2
  10. f(x)=ex,a=0
    1. The function is linear at y = 3 until it reaches (1, 3), at which point it increases as a line with slope 3.
    2. limh133hlimh1+3hh
    1. The function starts in the third quadrant as a straight line and passes through the origin with slope 2; then at (1, 2) it decreases convexly as 2/x.
    2. limh12hhlimh1+2x+h2xh
  11. 0
  12. 2x3
  13. f(x)=6x+2The function f(x) is graphed as an upward facing parabola with y intercept 4. The function f’(x) is graphed as a straight line with y intercept 2 and slope 6.
  14. f(x)=1(2x)3/2The function f(x) is in the first quadrant and has asymptotes at x = 0 and y = 0. The function f’(x) is in the fourth quadrant and has asymptotes at x = 0 and y = 0.
  15. f(x)=3x2The function f(x) starts is the graph of the cubic function shifted up by 1. The function f’(x) is the graph of a parabola that is slightly steeper than the normal squared function.
    1. Average rate at which customers spent on concessions in thousands per customer.
    2. Rate (in thousands per customer) at which x customers spent money on concessions in thousands per customer.
    1. Average grade received on the test with an average study time between two amounts.
    2. Rate (in percentage points per hour) at which the grade on the test increased or decreased for a given average study time of x hours.
    1. Average change of atmospheric pressure between two different altitudes.
    2. Rate (torr per foot) at which atmospheric pressure is increasing or decreasing at x feet.
    1. The rate (in degrees per foot) at which temperature is increasing or decreasing for a given height x.
    2. The rate of change of temperature as altitude changes at 1000 feet is -0.1 degrees per foot.
    1. The rate at which the number of people who have come down with the flu is changing t weeks after the initial outbreak.
    2. The rate is increasing sharply up to the third week, at which point it slows down and then becomes constant.
  16.  
  17. Time (seconds) h(t) (m/s)
    0 2
    1 2
    2 5.5
    3 10.5
    4 9.5
    5 7
  18.  
  19. G(t)=2.858t+0.0857This graph has the points (0, 0), (1, 2), (2, 4), (3, 13), (4, 25), and (5, 32). There is a quadratic line fit to the points with y intercept near 0.This graph has a straight line with y intercept near 0 and slope slightly less than 3.
  20.  
  21. H(t)=0,G(t)=2.858, and F(t)=1.222t+5.912 represent the acceleration of the rocket, with units of meters per second squared (m/s2).

Differentiation Rules

  1. f(x)=15x21
  2. f(x)=32x3+18x
  3. f(x)=270x4+39(x+1)2
  4. f(x)=5x2
  5. f(x)=4x4+2x22xx4
  6. f(x)=x218x+64(x27x+1)2
  7. The graph y is a slightly curving line with y intercept at 1. The line T(x) is straight with y intercept 3 and slope 1/2. T(x)=12x+3
  8. The graph y is a two crescents with the crescent in the third quadrant sloping gently from (−3, −1) to (−1, −5) and the other crescent sloping more sharply from (0.8, −5) to (3, 0.2). The straight line T(x) is drawn through (0, −5) with slope 4. T(x)=4x5
  9. h(x)=3x2f(x)+x3f(x)
  10. h(x)=3f(x)(g(x)+2)3f(x)g(x)(g(x)+2)2
  11. 169
  12. Undefined
    1. 23
    2. y=23x28
      The graph is a slightly deformed cubic function passing through the origin. The tangent line is drawn through (0, −28) with slope 23.
    1. 3
    2. y=3x+2The graph starts in the third quadrant, increases quickly and passes through the x axis near −0.9, then increases at a lower rate, passes through (0, 2), increases to (1, 5), and then decreases quickly and passes through the x axis near 1.2.
  13. y=7x3
  14. y=5x+7
  15. y=32x+152
  16. y=3x2+9x1
  17. 12121 or 0.0992 ft/s
    1. 2t42t3+200t+50(t3+50)2
    2. 0.02395 mg/L-hr, 0.01344 mg/L-hr, 0.003566 mg/L-hr, 0.001579 mg/L-hr
    3. The rate at which the concentration of drug in the bloodstream decreases is slowing to 0 as time increases.
    1. F(d)=2Gm1m2d3
    2. 1.33×107 N/m

Derivatives as Rates of Change

    1. v(t)=6t230t+36,a(t)=12t30
    2. Speeds up: (2,2.5)(3,); Slows down: (0,2)(2.5,3)
    1. 464ft/s2
    2. 32ft/s2
    1. 5 ft/s
    2. 9 ft/s
    1. 84 ft/s, −84 ft/s
    2. 84 ft/s
    3. 258 sec
    4. 32ft/s2 in both cases
    5. 18(25+965) sec
    6. 4965 ft/s
    1. Velocity is positive on (0,1.5)(6,7), negative on (1.5,2)(5,6), and zero on (2,5).
    2. The graph is a straight line from (0, 2) to (2, −1), then is discontinuous with a straight line from (2, 0) to (5, 0), and then is discontinuous with a straight line from (5, −4) to (7, 4).
    3. Acceleration is positive on (5,7), negative on (0,2), and zero on (2,5).
    4. The object is speeding up on (6,7)(1.5,2) and slowing down on (0,1.5)(5,6).
    1. R(x)=10x0.001x2
    2. R(x)=100.002x
    3. $6 per item, $0 per item
    1. C(x)=65
    2. R(x)=143x0.03x2,R(x)=1430.06x
    3. R(1000)=83,R(4000)=97. At a production level of 1000 cordless drills, revenue is increasing at a rate of $83 per drill; at a production level of 4000 cordless drills, revenue is decreasing at a rate of $97 per drill.
    4. P(x)=0.03x2+78x75000,P(x)=0.06x+78
    5. P(1000)=18,P(4000)=162. At a production level of 1000 cordless drills, profit is increasing at a rate of $18 per drill; at a production level of 4000 cordless drills, profit is decreasing at a rate of $162 per drill
    1. N(t)=3000(4t2+400(t2+100)2)
    2. N(0)=120,N(10)=0,N(20)=14.4,N(30)=9.6
    3. The bacteria population increases from time 0 to 10 hours; afterwards, the bacteria population decreases.
    4. N(0)=0,N(10)=6,N(20)=0.384,N(30)=0.432. The rate at which the bacteria is increasing is decreasing during the first 10 hours. Afterwards, the bacteria population is decreasing at a decreasing rate.