Introduction to Derivatives: Get Stronger Answer Key

Defining the Derivative

    1. [latex]4[/latex]
    2. [latex]8.5[/latex]
    3. [latex]-\frac{3}{4}[/latex]
    1. [latex]0.2[/latex]
    2. [latex]0.25[/latex]
    1. [latex]-4[/latex]
    2. [latex]y=3-4x[/latex]
    1. [latex]3[/latex]
    2. [latex]y=3x-1[/latex]
    1. [latex]\frac{-7}{9}[/latex]
    2. [latex]y=\frac{-7}{9}x+\frac{14}{3}[/latex]
    1. [latex]12[/latex]
    2. [latex]y=12x+14[/latex]
    1. [latex]-2[/latex]
    2. [latex]y=-2x-10[/latex]
  6. [latex]5[/latex]
  7. [latex]13[/latex]
  8. [latex]\frac{1}{4}[/latex]
  9. [latex]-\frac{1}{4}[/latex]
  10. [latex]3[/latex]
    1. (i) [latex]5.100000[/latex], (ii) [latex]5.010000[/latex], (iii) [latex]5.001000[/latex], (iv) [latex]5.000100[/latex], (v) [latex]5.000010[/latex], (vi) [latex]5.000001[/latex],
      (vii) [latex]4.900000[/latex], (viii) [latex]4.990000[/latex], (ix) [latex]4.999000[/latex], (x) [latex]4.999900[/latex], (xi) [latex]4.999990[/latex], (xii) [latex]4.999999[/latex]
    2. [latex]m_{\tan}=5[/latex]
    3. [latex]y=5x+3[/latex]
    1. (i) [latex]4.8771[/latex], (ii) [latex]4.9875[/latex], (iii) [latex]4.9988[/latex], (iv) [latex]4.9999[/latex], (v) [latex]4.9999[/latex], (vi) [latex]4.9999[/latex]
    2. [latex]m_{\tan}=5[/latex]
    3. [latex]y=5x+10[/latex]
    1. [latex]\frac{1}{3}[/latex]
    2. (i) [latex]0.\bar{3}[/latex] m/s, (ii) [latex]0.\bar{3}[/latex] m/s, (iii) [latex]0.\bar{3}[/latex] m/s, (iv) [latex]0.\bar{3}[/latex] m/s;
    3. [latex]0.\bar{3}=\frac{1}{3}[/latex] m/s
    1. [latex]2(h^2+6h+12)[/latex]
    2. (i) [latex]25.22[/latex] m/s, (ii) [latex]24.12[/latex] m/s, (iii) [latex]24.01[/latex] m/s, (iv) [latex]24[/latex] m/s
    3. [latex]24[/latex] m/s
  15. [latex]\underset{x\to 0^-}{\lim}\frac{x^{1/3}-0}{x-0}=\underset{x\to 0^-}{\lim}\frac{1}{x^{2/3}}=\infty[/latex]
  16. [latex]\underset{x\to 1^-}{\lim}\frac{1-1}{x-1}=0\ne 1=\underset{x\to 1^+}{\lim}\frac{x-1}{x-1}[/latex]
    1. (i) [latex]61.7244[/latex] ft/s, (ii) [latex]61.0725[/latex] ft/s, (iii) [latex]61.0072[/latex] ft/s, (iv) [latex]61.0007[/latex] ft/s
    2. At [latex]4[/latex] seconds the race car is traveling at a rate/velocity of [latex]61[/latex] ft/s.
    1. The vehicle represented by [latex]f(t)[/latex], because it has traveled [latex]2[/latex] feet, whereas [latex]g(t)[/latex] has traveled [latex]1[/latex] foot.
    2. The velocity of [latex]f(t)[/latex] [latex]is constant at [latex]1[/latex] ft/s, while the velocity of [latex]g(t)[/latex] is approximately [latex]2[/latex] ft/s.
    3. The vehicle represented by [latex]g(t)[/latex], with a velocity of approximately [latex]4[/latex] ft/s.
    4. Both have traveled [latex]4[/latex] feet in [latex]4[/latex] 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. [latex]a\approx -1.361, \, 2.694[/latex]
    1. [latex]N(x)=\frac{x}{30}[/latex]
    2. [latex]\sim 3.3[/latex] gallons. When the vehicle travels [latex]100[/latex] miles, it has used [latex]3.3[/latex] gallons of gas.
    3. [latex]\frac{1}{30}[/latex]. The rate of gas consumption in gallons per mile that the vehicle is achieving after having traveled [latex]100[/latex] miles.
    1. The function starts in the second quadrant and gently decreases, touches the origin, and then it increases gently.
    2. [latex]-0.028, \, -0.16, \, 0.16, \, 0.028[/latex]

The Derivative as a Function

  1. [latex]-3[/latex]
  2. [latex]8x[/latex]
  3. [latex]\frac{1}{\sqrt{2x}}[/latex]/li>
  4. [latex]\frac{-9}{x^2}[/latex]
  5. [latex]\frac{-1}{2x^{3/2}}[/latex]
  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. [latex]f(x)=3x^2+2, \, a=2[/latex]
  9. [latex]f(x)=x^4, \, a=2[/latex]
  10. [latex]f(x)=e^x, \, a=0[/latex]
    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. [latex]\underset{h\to 1^-}{\lim}\frac{3-3}{h}\ne \underset{h\to 1^+}{\lim}\frac{3h}{h}[/latex]
    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. [latex]\underset{h\to 1^-}{\lim}\frac{2h}{h}\ne \underset{h\to 1^+}{\lim}\frac{\frac{2}{x+h}-\frac{2}{x}}{h}[/latex]
  13. [latex]0[/latex]
  14. [latex]\frac{2}{x^3}[/latex]
  15. [latex]f^{\prime}(x)=6x+2[/latex]The 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.
  16. [latex]f^{\prime}(x)=-\frac{1}{(2x)^{3/2}}[/latex]The 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.
  17. [latex]f^{\prime}(x)=3x^2[/latex]The 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 [latex]x[/latex] 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 [latex]x[/latex] 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 [latex]x[/latex] feet.
    1. The rate (in degrees per foot) at which temperature is increasing or decreasing for a given height [latex]x[/latex].
    2. The rate of change of temperature as altitude changes at [latex]1000[/latex] feet is -[latex]0.1[/latex] degrees per foot.
    1. The rate at which the number of people who have come down with the flu is changing [latex]t[/latex] 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.
  23.  
  24. Time (seconds) [latex]h^{\prime}(t)[/latex] (m/s)
    [latex]0[/latex] [latex]2[/latex]
    [latex]1[/latex] [latex]2[/latex]
    [latex]2[/latex] [latex]5.5[/latex]
    [latex]3[/latex] [latex]10.5[/latex]
    [latex]4[/latex] [latex]9.5[/latex]
    [latex]5[/latex] [latex]7[/latex]
  25.  
  26. [latex]G^{\prime}(t)=2.858t+0.0857[/latex]This 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.
  27.  
  28. [latex]H''(t)=0, \, G''(t)=2.858[/latex], and [latex]F''(t)=1.222t+5.912[/latex] represent the acceleration of the rocket, with units of meters per second squared ([latex]\text{m/s}^2[/latex]).

Differentiation Rules

  1. [latex]f^{\prime}(x)=15x^2-1[/latex]
  2. [latex]f^{\prime}(x)=32x^3+18x[/latex]
  3. [latex]f^{\prime}(x)=270x^4+\frac{39}{(x+1)^2}[/latex]
  4. [latex]f^{\prime}(x)=\frac{-5}{x^2}[/latex]
  5. [latex]f^{\prime}(x)=\frac{4x^4+2x^2-2x}{x^4}[/latex]
  6. [latex]f^{\prime}(x)=\frac{−x^2-18x+64}{(x^2-7x+1)^2}[/latex]
  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. [latex]T(x)=\frac{1}{2}x+3[/latex]
  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. [latex]T(x)=4x-5[/latex]
  9. [latex]h^{\prime}(x)=3x^2f(x)+x^3f^{\prime}(x)[/latex]
  10. [latex]h^{\prime}(x)=\frac{3f^{\prime}(x)(g(x)+2)-3f(x)g^{\prime}(x)}{(g(x)+2)^2}[/latex]
  11. [latex]\frac{16}{9}[/latex]
  12. Undefined
    1. [latex]23[/latex]
    2. [latex]y=23x-28[/latex]
      The graph is a slightly deformed cubic function passing through the origin. The tangent line is drawn through (0, −28) with slope 23.
    1. [latex]3[/latex]
    2. [latex]y=3x+2[/latex]The 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.
  15. [latex]y=-7x-3[/latex]
  16. [latex]y=-5x+7[/latex]
  17. [latex]y=-\frac{3}{2}x+\frac{15}{2}[/latex]
  18. [latex]y=-3x^2+9x-1[/latex]
  19. [latex]\frac{12}{121}[/latex] or [latex]0.0992[/latex] ft/s
    1. [latex]\frac{-2t^4-2t^3+200t+50}{(t^3+50)^2}[/latex]
    2. [latex]-0.02395[/latex] mg/L-hr, [latex]−0.01344[/latex] mg/L-hr, [latex]−0.003566[/latex] mg/L-hr, [latex]−0.001579[/latex] mg/L-hr
    3. The rate at which the concentration of drug in the bloodstream decreases is slowing to [latex]0[/latex] as time increases.
    1. [latex]F^{\prime}(d)=\frac{-2Gm_1 m_2}{d^3}[/latex]
    2. [latex]-1.33 \times 10^{-7}[/latex] N/m

Derivatives as Rates of Change

    1. [latex]v(t)=6t^2-30t+36, \, a(t)=12t-30[/latex]
    2. Speeds up: [latex](2,2.5)\cup (3,\infty)[/latex]; Slows down: [latex](0,2)\cup (2.5,3)[/latex]
    1. [latex]464 \, \text{ft/s}^2[/latex]
    2. [latex]-32 \, \text{ft/s}^2[/latex]
    1. [latex]5[/latex] ft/s
    2. [latex]9[/latex] ft/s
    1. [latex]84[/latex] ft/s, −[latex]84[/latex] ft/s
    2. [latex]84[/latex] ft/s
    3. [latex]\frac{25}{8}[/latex] sec
    4. [latex]-32 \, \text{ft/s}^2[/latex] in both cases
    5. [latex]\frac{1}{8}(25+\sqrt{965})[/latex] sec
    6. [latex]-4\sqrt{965}[/latex] ft/s
    1. Velocity is positive on [latex](0,1.5)\cup (6,7)[/latex], negative on [latex](1.5,2)\cup (5,6)[/latex], and zero on [latex](2,5)[/latex].
    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 [latex](5,7)[/latex], negative on [latex](0,2)[/latex], and zero on [latex](2,5)[/latex].
    4. The object is speeding up on [latex](6,7)\cup (1.5,2)[/latex] and slowing down on [latex](0,1.5)\cup (5,6)[/latex].
    1. [latex]R(x)=10x-0.001x^2[/latex]
    2. [latex]R^{\prime}(x)=10-0.002x[/latex]
    3. [latex]$6[/latex] per item, [latex]$0[/latex] per item
    1. [latex]C^{\prime}(x)=65[/latex]
    2. [latex]R(x)=143x-0.03x^2, \, R^{\prime}(x)=143-0.06x[/latex]
    3. [latex]R^{\prime}(1000)=83, \, R^{\prime}(4000)=-97[/latex]. At a production level of [latex]1000[/latex] cordless drills, revenue is increasing at a rate of [latex]$83[/latex] per drill; at a production level of 4000 cordless drills, revenue is decreasing at a rate of [latex]$97[/latex] per drill.
    4. [latex]P(x)=-0.03x^2+78x-75000, \, P^{\prime}(x)=-0.06x+78[/latex]
    5. [latex]P^{\prime}(1000)=18, \, P^{\prime}(4000)=-162[/latex]. At a production level of [latex]1000[/latex] cordless drills, profit is increasing at a rate of [latex]$18[/latex] per drill; at a production level of 4000 cordless drills, profit is decreasing at a rate of [latex]$162[/latex] per drill
    1. [latex]N^{\prime}(t)=3000(\frac{-4t^2+400}{(t^2+100)^2})[/latex]
    2. [latex]N^{\prime}(0)=120, \, N^{\prime}(10)=0, \, N^{\prime}(20)=-14.4, \, N^{\prime}(30)=-9.6[/latex]
    3. The bacteria population increases from time [latex]0[/latex] to [latex]10[/latex] hours; afterwards, the bacteria population decreases.
    4. [latex]N''(0)=0, \, N''(10)=-6, \, N''(20)=0.384, \, N''(30)=0.432[/latex]. The rate at which the bacteria is increasing is decreasing during the first [latex]10[/latex] hours. Afterwards, the bacteria population is decreasing at a decreasing rate.