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Applied Calculus

Kevin Gonzales, Eric Hopkins, Catherine Zimmitti, Cheryl Kane, Modified to fit Applied Calculus from Coordinated Calculus by Nathan Wakefield et. al., Based upon Active Calculus by Matthew Boelkins

Appendix B Answers to Selected Exercises

This appendix contains answers to all non- WeBWorK exercises in the text. For WeBWorK exercises, please use the HTML version of the text for access to answers and solutions.

0 PreCalculus Review
0.1 Functions
0.1.11 Exercises

0.1.11.1. Slope and Intercept.

Answer 1.
\(2\)
Answer 2.
\(\frac{-11}{3}\)

0.1.11.2. Graphs of Linear Equations.

Answer 1.
VI
Answer 2.
II
Answer 3.
I
Answer 4.
III
Answer 5.
IV
Answer 6.
V

0.1.11.3. Proportionality.

Answer 1.
\(h\)
Answer 2.
\(kh^{2}\)

0.1.11.4. Finding Lines.

Answer.
\(\frac{2-4}{3--3}\!\left(x+3\right)+4\)

1 Limits
1.1 Introduction to Limits
1.1.3 Exercises

1.1.3.1. Limits on a piecewise graph.

Answer 1.
\(5\)
Answer 2.
\(9\)
Answer 3.
\(\text{none}\)
Answer 4.
\(-4\)

1.1.3.2. Estimating a limit numerically.

Answer.
\(9\)

1.1.3.3. Limits for a piecewise formula.

Answer 1.
\(4\cdot 4+-5\)
Answer 2.
\(4\cdot 4-5\)
Answer 3.
\(4\cdot 4-5\)

1.1.3.4. Calculating Limits of Rational Functions.

Answer.
\(0.5\)

1.1.3.5. One-Sided Limits.

Answer 1.
\(6\)
Answer 2.
\(-3\)
Answer 3.
\(\text{DNE}\)

1.2 Algebraic Limits
1.2.4 Exercises

1.2.4.1. Limits on a piecewise graph.

Answer 1.
\(2\)
Answer 2.
\(6\)
Answer 3.
\(\text{none}\)
Answer 4.
\(-1\)

1.2.4.2. Limits for a piecewise formula.

Answer 1.
\(3\cdot 3+-3\)
Answer 2.
\(3\cdot 3-3\)
Answer 3.
\(3\cdot 3-3\)

1.2.4.3. Calculating Limits of Rational Functions.

Answer.
\(0.2\)

1.2.4.4. One-Sided Limits.

Answer 1.
\(16\)
Answer 2.
\(0\)
Answer 3.
\(\text{DNE}\)

1.2.4.5. Evaluating a limit algebraically.

Answer.
\(-14\)

1.3 Limits to Infinity
1.3.3 Exercises

1.3.3.1. Calculating Limits of Rational Functions.

Answer.
\(0.125\)

1.3.3.2. Evaluating a limit algebraically.

Answer.
\(-16\)

1.4 Continuous Functions
1.4.6 Exercises

1.4.6.1. Types of discontinuity.

Answer.
\(2\)

1.4.6.2. Types of discontinuity.

Answer.
\(1\)

1.4.6.4. Determining continuity from a graph.

Answer 1.
is
Answer 2.
is not
Answer 3.
is
Answer 4.
is not
Answer 5.
is not
Answer 6.
is

1.4.6.5. Determining continuity from a graph.

Answer 1.
is not
Answer 2.
is
Answer 3.
is not

1.4.6.6. Interpretting continuity.

Answer 1.
\(7\)
Answer 2.
\(6\)
Answer 3.
\(7\)
Answer 4.
\(21\)
Answer 5.
\(7\)

1.4.6.7. Values that make a function continuous.

Answer.
\(5\cdot 4\)

1.4.6.8. Values that make a function continuous.

Answer.
\(\frac{-15}{8}\)

1.4.6.9. Values that make a function continuous.

Answer.
\(8\cdot 2^{4}\)

2 Derivatives
2.1 The Average Rate of Change
2.1.3 Exercises

2.1.3.1. Estimating derivative values graphically.

Answer 1.
\(0.905829\)
Answer 2.
\(-0.5\)
Answer 3.
\(3.22474\)
Answer 4.
\(0.355567\)
Answer 5.
\(2.25\)

2.1.3.2. Tangent line to a curve.

Answer 1.
\(1.2\)
Answer 2.
\(8\)
Answer 3.
\(1.2\)
Answer 4.
\(\frac{-1\cdot \left(-0.05\right)}{0.03}\)

2.1.3.3. Interpreting values and slopes from a graph.

Answer 1.
\({\verb!<!}\)
Answer 2.
\({\verb!>!}\)
Answer 3.
\({\verb!<!}\)
Answer 4.
\({\verb!<!}\)

2.1.3.4. Estimating a derivative value graphically.

Answer.
\(36\)

2.1.3.5. Estimating a derivative from the limit definition.

Answer.
\(354.9\)

2.1.3.6. Using a graph.

Answer.
  2. \(AV_{[-3,-1]} \approx 1.15\text{;}\) \(AV_{[0,2]} \approx -0.4\text{.}\)
  3. \(f'(-3) \approx 3\text{;}\) \(f'(0) \approx -\frac{1}{2}\text{.}\)

2.1.3.7. Creating graphs with certain properties.

Answer.
  1. For instance, you could let \(f(-3) = 3\) and have \(f\) pass through the points \((-3,3)\text{,}\) \((-1,-2)\text{,}\) \((0,-3)\text{,}\) \((1,-2)\text{,}\) and \((3,-1)\) and draw the desired tangent lines accordingly.
  2. For instance, you could draw a function \(g\) that passes through the points \((-2,3)\text{,}\) \((-1,2)\text{,}\) \((1,0)\text{,}\) \((2,0)\text{,}\) and \((3,3)\) in such a way that the tangent line at \((-1,2)\) is horizontal and the tangent line at \((2,0)\) has slope \(1\text{.}\)

2.1.3.8. Population Growth.

Answer.
  1. \(AV_{[0,7]}=\frac{0.1175}{7} \approx 0.01679\) billion people per year; \(P'(7) \approx 0.1762\) billion people per year; \(P'(7) \gt AV_{[0,7]}\text{.}\)
  2. \(AV_{[19,29]} \approx 0.02234\) billion people/year.
  3. We will say that today’s date is July 1, 2015, which means that \(t = 22.5\text{;}\)
    \begin{equation*} P'(22.5) = \lim_{h \to 0} \frac{115(1.014)^{22.5+h}-115(1.014)^{22.5}}{h}; \end{equation*}
    \(P'(22.5) \approx 0.02186\) billions of people per year.
  4. \(y - 1.57236 = 0.02186(t-22.5)\text{.}\)

2.1.3.9. Using the limit definition of the derivative.

Answer.
  1. All three approaches show that \(f'(2) = 1\text{.}\)
  2. All three approaches show that \(f'(1) = -1\text{.}\)
  3. All three approaches show that \(f'(1) = \frac{1}{2}\text{.}\)
  4. All three approaches show that \(f'(1)\) does not exist.
  5. The first two approaches show that \(f'(\frac{\pi}{2}) = 0\text{.}\)

2.2 The Derivative Function
2.2.5 Exercises

2.2.5.1. Applying the limit definition of the derivative.

Answer 1.
\(4\!\left(x+h\right)^{2}-6-\left(4x^{2}-6\right)\)
Answer 2.
\(4\cdot 2x\)

2.2.5.2. Sketching the derivative.

Answer.
\(\text{7}\)

2.2.5.3. Comparing function and derivative values.

Answer 1.
\(\text{x6}\)
Answer 2.
\(\text{x4}\)
Answer 3.
\(\text{x5}\)
Answer 4.
\(\text{x3}\)

2.2.5.4. Limit definition of the derivative for a rational function.

Answer 1.
\(-1\)
Answer 2.
\(-1\)
Answer 3.
\(\frac{-1}{4}\)
Answer 4.
\(\frac{-1}{16}\)

2.2.5.5. Derivative from Tangent Line.

Answer 1.
\(-5\)
Answer 2.
\(\frac{9--5}{-7--8}\)

2.2.5.6. Determining functions from their derivatives.

Answer.
  1. See the figure below.
  2. See the figure below.
  3. One example of a formula for \(f\) is \(f(x) = \frac{1}{2}x^2 - 1\text{.}\)

2.2.5.7. Algebraic and graphical connections between a function and its derivative.

Answer.
  1. \(g'(x) = 2x - 1\text{.}\)
  3. \(p'(x) = 10x - 4\text{.}\)
  4. The constants \(3\) and \(12\) don’t seem to affect the results at all. The coefficient \(-4\) on the linear term in \(p(x)\) appears to make the ``\(-4\)’’ appear in \(p'(x)= 10x - 4\text{.}\) The leading coefficient \(5\) in \((x) = 5x^2 - 4x + 12\) leads to the coefficient of ``\(10\)’’ in \(p'(x) = 10x -4\text{.}\)

2.2.5.8. Graphing functions based on continuity and derivatives.

Answer.
  1. \(g\) is linear.
  2. On \(-3.5 \lt x \lt -2\text{,}\) \(-2 \lt x \lt 0\) and \(2 \lt x \lt 3.5\text{.}\)
  3. At \(x = -2, 0, 2\text{;}\) \(g\) must have sharp corners at these points.

2.2.5.9. Graphing the Derivative Function.

Answer.

2.3 Differentiability
2.3.5 Exercises

2.3.5.1. Continuity and differentiability of a graph.

Answer 1.
\(3\)
Answer 2.
\(3\)

2.3.5.4. Continuity and differentiability of a graph.

Answer.
  1. \(a = 0\text{.}\)
  2. \(a = 0, 3\text{.}\)
  3. \(a = -2, 0, 1, 2, 3\text{.}\)

2.3.5.5. Examples of functions.

Answer.
  1. \(f(x) = |x-2|\text{.}\)
  2. Impossible.
  3. Let \(f\) be the function defined to be \(f(x) = 1\) for every value of \(x \ne -2\text{,}\) and such that \(f(-2) = 4\text{.}\)

2.3.5.6. Estimating the derivative at a point.

Answer.
  1. At \(x = 0\text{.}\)
    \begin{align*} g'(0) & = \lim_{h \to 0} \frac{g(0+h) - g(0)}{h}\\ & = \lim_{h \to 0} \frac{\sqrt{|h|} - \sqrt{|0|}}{h}\\ & = \lim_{h \to 0} \frac{\sqrt{|h|}}{h} \end{align*}
  2. \(h\) \(0.1\) \(0.01\) \(0.001\) \(0.0001\) \(-0.1\) \(-0.01\) \(-0.001\) \(-0.0001\)
    \(\sqrt{|h|}/h\) \(3.162\) \(10\) \(31.62\) \(100\) \(-3.162\) \(-10\) \(-31.62\) \(-100\)
    \(g'(0)\) does not exist.

2.4 Derivative Rules
2.4.5 Exercises

2.4.5.1. Derivative of a power function.

Answer.
\(\frac{11}{12}x^{\frac{11}{12}-1}\)

2.4.5.2. Derivative of a rational function.

Answer.
\(-1\cdot 13x^{-1\cdot \left(13+1\right)}\)

2.4.5.3. Derivative of a root function.

Answer.
\(\frac{1}{8}x^{\frac{1}{8}-1}\)

2.4.5.4. Derivative of a quadratic function.

Answer.
\(2\cdot 8t-8\)

2.4.5.5. Derivative of a sum of power functions.

Answer.
\(15\cdot 8t^{8-1}-\frac{9}{2}t^{\frac{-1}{2}}-\frac{11}{t^{2}}\)

2.4.5.6. Simplifying a product before differentiating.

Answer.
\(\frac{2\cdot 1+1}{2}x^{\frac{2\cdot 1-1}{2}}+\frac{2}{2}x^{\frac{-1}{2}}\)

2.4.5.7. Simplifying a quotient before differentiating.

Answer.
\(\frac{3x^{2}x-\left(x^{3}+12\right)}{x^{2}}\)

2.4.5.8. Determining where \(f'(x) = 0\).

Answer.
\(4, -6\)

2.5 The Tangent Line
2.5.4 Exercises

2.5.4.1. Sign of tangent line slope.

Answer 1.
\(0\)
Answer 2.
\(0\)
Answer 3.
\(-3, 0, 3\)

2.5.4.2. Sign of tangent line slope.

Answer 1.
\(0\)
Answer 2.
\(0\)
Answer 3.
\(1, 3\)

2.5.4.3. Equation of tangent line for polynomial.

Answer 1.
\(6\cdot 3x^{2}-3\cdot 2x-7\)
Answer 2.
\(77\)
Answer 3.
\(77\!\left(x-\left(-2\right)\right)+\left(-55\right)\)

2.5.4.4. Equation of tangent line for polynomial.

Answer 1.
\(7\cdot 4x^{3}-2x\)
Answer 2.
\(9590\)
Answer 3.
\(9590\!\left(x-7\right)+16757\)

2.5.4.5. Equation of tangent line for non-polynomial.

Answer 1.
\(8\cdot 1.5x^{0.5}-5\frac{1}{2\sqrt{x}}\)
Answer 2.
\(19.3412\)
Answer 3.
\(19.3412\!\left(x-3\right)+37.909\)

2.5.4.6. Equation of tangent line for non-polynomial.

Answer.
\(-0.888889\!\left(x-\left(-3\right)\right)+\left(-3.16667\right)\)

2.5.4.7. Equation of tangent line.

Answer.
\(-11\!\left(x-1\right)+11.3333\)

2.6 The Product and Quotient Rules
2.6.5 Exercises

2.6.5.1. Derivative of a quotient of linear functions.

Answer.
\(\frac{3\!\left(7t+2\right)-7\!\left(3t+10\right)}{\left(7t+2\right)^{2}}\)

2.6.5.2. Derivative of a rational function.

Answer.
\(\frac{4r^{4-1}\!\left(6r+8\right)-6r^{4}}{\left(6r+8\right)^{2}}\)

2.6.5.3. Product and quotient rules with graphs.

Answer 1.
\(0.666667\cdot 0+-0.666667\cdot 1.33333\)
Answer 2.
\(\frac{-4\cdot \left(-1\right)--1\cdot \left(-2\right)}{\left(-1\right)^{2}}\)

2.6.5.4. Product and quotient rules with given function values.

Answer 1.
\(3\cdot 1+2\cdot 5\)
Answer 2.
\(\frac{2\cdot 5-3\cdot 1}{5\cdot 5}\)

2.6.5.5. Product rule.

Answer.
\(\left(2t+5\right)\!\left(6t^{-2}+4t^{-3}\right)+\left(t^{2}+5t+3\right)\!\left(-2\cdot 6t^{-3}-3\cdot 4t^{-4}\right)\)

2.6.5.6. Product rule and tangent lines.

Answer.
\(288\!\left(x-2\right)+-93\)

2.6.5.7. Quotient rule and tangent lines.

Answer.
\(-0.177515x+0.961538--0.177515\cdot 5\)

2.6.5.8. Tangent lines using given function values.

Answer.
  1. \(h(2) = -15\text{;}\) \(h'(2) = 23/2\text{.}\)
  2. \(L(x) = -15 + 23/2(x-2)\text{.}\)
  3. Increasing.
  4. \(r(2.06) \approx -0.5796\text{.}\)

2.6.5.9. Product and quotient rule with non-basic functions.

Answer.
  1. \(w'(t) = t^t\left(\ln t+1\right)\cdot\left(\arccos t\right)+t^t\cdot\frac{-1}{\sqrt{1-t^2}} \text{.}\)
  2. \(L(t) \approx 0.740-0.589(t-0.5)\text{.}\)
  3. Increasing.

2.6.5.10. Product and quotient rules with analysis of graphs.

Answer.
  1. \(r'(-2) = 2\) and \(r'(0) = 0.25\text{.}\)
  2. At \(x = -1\) and \(x = 1\text{.}\)
  3. \(L(x) = 2\text{.}\)
  4. \(z'(0) = \frac{1}{16}\) and \(z'(2) = -1\text{.}\)
  5. At \(x = -1\text{,}\) \(x = 1\text{,}\) \(x = -1.5\text{,}\) and \(x = 1\text{.}\)

2.6.5.11. An application to crop yield.

Answer.
  1. \(C(t) = A(t)Y(t)\) bushels in year \(t\text{.}\)
  2. \(1 190 000\) bushels of corn.
  3. \(C'(t) = A(t)Y'(t) + A'(t)Y(t)\text{.}\)
  4. \(C'(0) = 158 000\) bushels per year.
  5. \(C(1) \approx 1 348 000\)bushels.

2.6.5.12. An application to fuel consumption.

Answer.
  1. \(g(80) = 20\) kilometers per liter, and \(g'(80) = -0.16\text{.}\) kilometers per liter per kilometer per hour.
  2. \(h(80) = 4\) liters per hour and \(h'(80) = 0.082\) liters per hour per kilometer per hour.
  3. Think carefully about units and how each of the three pairs of values expresses fundamentally the same facts.

2.7 The Chain Rule
2.7.6 Exercises

2.7.6.1. Chain rule with graphs.

Answer 1.
\(-0.5\)
Answer 2.
\(-0.5\)
Answer 3.
\(-0.5\)

2.7.6.2. Chain rule with function values.

Answer 1.
\(4\)
Answer 2.
\(4\cdot 1\)
Answer 3.
\(4\)
Answer 4.
\(3\cdot 1\)
Answer 5.
\(\frac{1\cdot 4-3\cdot 1}{4^{2}}\)

2.7.6.4. Evaluating the derivative using chain rule.

Answer 1.
\(-3\!\left(3x+3\right)^{-4}\cdot 3\)
Answer 2.
\(-3\cdot \left(3\cdot 5+3\right)^{-3-1}\cdot 3\)

2.7.6.8. Using the chain rule to compare composite functions.

Answer.
  1. \(h'\left( \frac{\pi}{4} \right) = \frac{3}{2\sqrt{2}}\text{.}\)
  2. \(r'(0.25) = \cos(0.25^3) \cdot 3(0.25)^2 \approx 0.1875 \gt h'(0.25) = 3\sin^2(0.25) \cdot \cos(0.25) \approx 0.1779\text{;}\) \(r\) is changing more rapidly.
  3. \(h'(x)\) is periodic; \(r'(x)\) is not.

2.7.6.9. More on using the chain rule with graphs.

Answer.
  1. \(C'(0) = 0\) and \(C'(3) = -\frac{1}{2}\text{.}\)
  2. Consider \(C'(1)\text{.}\) By the chain rule, we’d expect that \(C'(1) = p'(q(1)) \cdot q'(1)\text{,}\) but we know that \(q'(1)\) does not exist since \(q\) has a corner point at \(x = 1\text{.}\) This means that \(C'(1)\) does not exist either.
  3. Since \(Y(x) = q(q(x))\text{,}\) the chain rule implies that \(Y'(x) = q'(q(x)) \cdot q'(x)\text{,}\) and thus \(Y'(-2) = q'(q(-2)) \cdot q'(-2) = q'(-1) \cdot q'(-2)\text{.}\) But \(q'(-1)\) does not exist, so \(Y'(-2)\) also fails to exist. Using \(Z(x) = q(p(x))\) and the chain rule, we have \(Z'(x) = q'(p(x)) \cdot p'(x)\text{.}\) Therefore \(Z'(0) = q'(p(0)) \cdot p'(0) = q'(-0.5) \cdot p'(0) = 0 \cdot 0.5 = 0\text{.}\)

2.7.6.10. Applying the chain rule in a physical context.

Answer.
  1. \(\frac{dV}{dh} = \pi \left(8h-h^2 \right)\text{,}\) cubic feet per foot.
  2. \(\frac{dV}{dt} = \frac{\pi}{3}\left[ 12 \cdot 2 (\sin(\pi t) + 1) \cdot \pi \cos(\pi t) - 3 (\sin(\pi t) + 1)^2 \cdot \pi \cos(\pi t) \right]\) cubic feet per hour.
  3. \(\left. \frac{dV}{dt} \right|_{t=0} = 7 \pi^2\) cubic feet per hour.
  4. In (a) we are determining the instantaneous rate at which the volume changes as we increase the height of the water in the tank, while in (c) we are finding the instantaneous rate at which volume changes as we increase time.

2.8 Higher-Order Derivatives
2.8.4 Exercises

2.8.4.1. Comparing \(f, f', f''\) values.

Answer 1.
\(\text{negative}\)
Answer 2.
\(\text{positive}\)
Answer 3.
\(\text{positive}\)

2.8.4.2. Signs of \(f, f', f''\) values.

Answer 1.
\(\text{positive}\)
Answer 2.
\(\text{positive}\)
Answer 3.
\(\text{negative}\)
Answer 4.
\(\text{negative}\)
Answer 5.
\(\text{negative}\)
Answer 6.
\(\text{positive}\)
Answer 7.
\(\text{zero}\)
Answer 8.
\(\text{negative}\)
Answer 9.
\(\text{zero}\)
Answer 10.
\(\text{positive}\)
Answer 11.
\(\text{negative}\)
Answer 12.
\(\text{negative}\)
Answer 13.
\(\text{positive}\)
Answer 14.
\(\text{positive}\)
Answer 15.
\(\text{positive}\)

2.8.4.3. Acceleration from velocity.

Answer 1.
\(34.09\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s^{2}}}\)
Answer 2.
\(25\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s^{2}}}\)

2.8.4.4. Rates of change of stock values.

Answer 1.
\(\text{positive}\)
Answer 2.
\(\text{positive}\)
Answer 3.
\(\text{positive}\)
Answer 4.
\(\text{positive}\)

2.8.4.5. Interpreting a graph of \(f'\).

Answer 1.
\(\text{x6}\)
Answer 2.
\(\text{x1}\)
Answer 3.
\(\text{x4}\)
Answer 4.
\(\text{x2}\)
Answer 5.
\(\text{x3}\)
Answer 6.
\(\text{x1}\)

2.8.4.6. Higher order derivatives.

Answer 1.
\(24x^{7}-6x^{5}+4x^{3}\)
Answer 2.
\(168x^{6}-30x^{4}+12x^{2}\)
Answer 3.
\(1008x^{5}-120x^{3}+24x\)
Answer 4.
\(5040x^{4}-360x^{2}+24\)

2.8.4.7. Chain rule with higher order derivatives.

Answer 1.
\(\frac{9t}{\sqrt{9t^{2}+11}}\)
Answer 2.
\(\frac{99}{\left(9t^{2}+11\right)^{1.5}}\)

2.8.4.8. Quotient rule with higher order derivatives.

Answer 1.
\(\frac{36}{\left(4-5x\right)^{2}}\)
Answer 2.
\(\frac{360}{\left(4-5x\right)^{3}}\)

2.8.4.9. Computing velocity and acceleration.

Answer 1.
\(4t^{3}-196t\)
Answer 2.
\(12t^{2}-196\)
Answer 3.
\(1600\)
Answer 4.
\(-480\)
Answer 5.
\(-88\)

2.8.4.10. Interpretting a graph of \(f\) based on the first and second derivatives.

Answer.
  1. \(f\) is increasing and concave down at \(x=2\text{.}\)
  2. Greater.
  3. Less.

2.8.4.11. Interpreting a graph of \(f'\).

Answer.
  1. \(g'(2) \approx 1.4\text{.}\)
  2. At most one.
  3. \(9\text{.}\)
  4. \(g''(2) \approx 5.5 \text{.}\)

2.8.4.12. Using data to interpret derivatives.

Answer.
  1. \(h'(4.5) \approx 14.3\text{;}\) \(h'(5) \approx 21.2\text{;}\) \(h'(5.5) \approx = 23.9\text{;}\) rising most rapidly at \(t = 5.5\text{.}\)
  2. \(h'(5) \approx 9.6 \text{.}\)
  3. Acceleration of the bungee jumper in feet per second per second.
  4. \(0 \lt t \lt 2\text{,}\) \(6 \lt t \lt 10\text{.}\)

2.8.4.13. Sketching functions.

Answer.

3 Exponentials and Logarithms
3.1 Exponential Functions
3.1.4 Exercises

3.1.4.1. General Exponential Functions.

Answer 1.
\(8\)
Answer 2.
\(e^{2}\)
Answer 3.
\(\text{growth}\)

3.1.4.4. Half-Life.

Answer.
\(878.371\ {\rm hr}\)

3.1.4.5. Applied Half-Life.

Answer 1.
\(\text{No}\)
Answer 2.
\(346.076\)

3.1.4.7. Mixing rules: chain, product, sum.

Answer.
\(5e^{5x}\!\left(x^{2}+6^{x}\right)+e^{5x}\!\left(2x+\ln\!\left(6\right)\cdot 6^{x}\right)\)

3.1.4.8. Mixing rules: chain and product.

Answer.
\(6t^{6-1}e^{-1ct}-ct^{6}e^{-1ct}\)

3.1.4.9. Using the chain rule repeatedly.

Answer.
\(\frac{-1\cdot 5te^{-1\cdot 5t^{2}}}{\sqrt{e^{-1\cdot 5t^{2}}+8}}\)

3.3 Derivatives of Log Functions
3.3.3 Exercises

3.3.3.1. Composite function involving logarithms and polynomials.

Answer.
\(\frac{9t^{8}}{t^{9}+7}\)

3.3.3.2. Composite function from a graph.

Answer 1.
\(\frac{4\cdot 4^{4-1}\!\left(4.4-4\right)}{1.3-1}\)
Answer 2.
\(\frac{1.3-1}{4.4-4}\)

3.3.3.3. Derivatives and properties of logs.

Answer 1.
\(\frac{3}{x}\)
Answer 2.
\(\frac{3}{x}\)

3.3.3.4. Derivatives using properties of logs.

Answer.
\(\frac{\frac{4}{4x+\left(-7\right)}-\frac{9}{9x+6}}{2}\)

3.3.3.5. Tangent lines using logs.

Answer.
\(0.75\!\left(x-4\right)+3\ln\!\left(4\right)\)

4 Applications of the Derivative
4.1 First Derivative Test
4.1.5 Exercises

4.1.5.1. Increasing and decreasing.

Answer 1.
\(\left(-14,-8\right)\cup \left(-2,4\right)\)
Answer 2.
\(\left(-8,-2\right)\)
Answer 3.
\(-14, -8, -2, 4\)
Answer 4.
\(\left(-14,-11\right)\cup \left(-5,1\right)\)
Answer 5.
\(\left(-11,-5\right)\cup \left(1,4\right)\)

4.1.5.2. Finding critical values.

Answer.
\(\frac{4}{5}\)

4.1.5.5. Identifying relative extrema.

Answer 1.
\(1, -2\)
Answer 2.
\(0\)

4.1.5.6. Determining intervals of increasing and decreasing and finding extrema.

Answer 1.
\(\left(-\infty ,1\right)\cup \left(5,\infty \right)\)
Answer 2.
\(\left(1,5\right)\)
Answer 3.
\(1\)
Answer 4.
\(5\)

4.2 Second Derivative Test
4.2.4 Exercises

4.2.4.2. Finding critical points and inflection points.

Answer 1.
\(0\)
Answer 2.
\(-0.447213595499958, 0.447213595499958\)
Answer 3.
\(0\)
Answer 4.
\(-0.447213595499958, 0.447213595499958\)

4.2.4.3. Finding inflection points.

Answer.
\(-7, 0.125\)

4.2.4.4. Matching graphs of \(f,f',f''\).

Answer 1.
B
Answer 2.
C
Answer 3.
A

4.2.4.7. Using a derivative graph to analyze a function.

Answer.
  1. \(f'\) is positive for \(-1 \lt x lt 1\) and for \(x \gt 1\text{;}\) \(f'\) is negative for all \(x \lt -1\text{.}\) \(f\) has a local minimum at \(x = -1\text{.}\)
  2. A possible graph of \(y = f''(x)\) is shown at right in the figure.
  3. \(f''(x)\) is negative for \(-0.35 \lt x \lt 1\text{;}\) \(f''(x)\) is positive everywhere else; \(f\) has points of inflection at \(x \approx -0.35\) and \(x = 1\text{.}\)
  4. A possible graph of \(y = f(x)\) is shown at left in the figure.

4.2.4.8. Using derivative tests.

Answer.
  1. Neither.
  2. \(g''(2) = 0\text{;}\) \(g''\) is negative for \(1 \lt x \lt 2\) and positive for \(2 \lt x \lt 3\text{.}\)
  3. \(g\) has a point of inflection at \(x = 2\text{.}\)

4.2.4.9. Using a derivative graph to analyze a function.

Answer.
  1. \(h\) can have no, one, or two real zeros.
  2. One root is negative and the other positive.
  3. \(h\) will look like a line with slope \(3\text{.}\)
  4. \(h\) is concave up everywhere; \(h\) is almost linear for large values of \(|x|\text{.}\)

4.2.4.10. Applying derivative tests.

Answer.
  1. \(p''(x)\) is negative for \(-1 \lt x \lt 2\) and positive for all other values of \(x\text{;}\) \(p\) has points of inflection at \(x = -1\) and \(x = 2\text{.}\)
  2. Local maximum.
  3. Neither.

4.3 Absolute Extrema
4.3.3 Exercises

4.3.3.1. Finding absolute Extrema.

Answer 1.
\(-122\)
Answer 2.
\(1606\)

4.3.3.2. Finding absolute Extrema.

Answer 1.
\(-80\)
Answer 2.
\(1520\)

4.3.3.3. Analyzing Function Behavior.

Answer 1.
\(-1\)
Answer 2.
\(\left(-\infty ,-1\right)\)
Answer 3.
\(\left(-1,\infty \right)\)
Answer 4.
\(5\)
Answer 5.
\(2\)
Answer 6.
\(2\)
Answer 7.
Undefined

4.3.3.5. Conditions for When absolute Extrema May Occur.

Answer.
  1. Not enough information is given.
  2. absolute minimum at \(x = a\text{.}\)
  3. absolute minimum at \(x = a\text{;}\) absolute maximum at \(x = b\text{.}\)
  4. Not enough information is provided.

4.3.3.6. Finding Extrema on Closed and Bounded Intervals.

Answer.
  1. absolute maximum \(p(0) = p(a) = 0\text{;}\) absolute minimum \(p\left( \frac{a}{\sqrt{3}} \right) = -\frac{2a^3}{3\sqrt{3}}\text{.}\)
  2. absolute max \(r\left( \frac{1}{b} \right) \approx 0.368 \frac{a}{b}\text{;}\) absolute min \(r\left( \frac{2}{b} \right) \approx 0.270 \frac{a}{b}\text{.}\)
  3. absolute minimum \(g(b) = a(1-e^{-b^2})\text{;}\) absolute maximum \(g(3b) = a(1-e^{-3b^2})\text{.}\)

4.3.3.7. Conditions for Where absolute Extrema May Occur.

Answer.
  1. absolute maximum at \(x=a\text{;}\) absolute minimum at \(x=b\text{.}\)
  2. absolute maximum at \(x=c\text{;}\) absolute minimum at either \(x=a\) or \(x=b\text{.}\)
  3. absolute minimum at \(x=a\) and \(x=b\text{;}\) absolute maximum somewhere in \((a,b)\text{.}\)
  4. absolute minimum at \(x=c\text{;}\) absolute maximum value at \(x = a\text{.}\)

4.4 Applied Optimization
4.4.5 Exercises

4.4.5.1. Maximizing the volume of a box.

Answer 1.
\(10.7218\times4.72184\)
Answer 2.
\(1.63908\)

4.4.5.2. Minimizing the cost of a container.

Answer.
\(\$321.85\)

4.4.5.3. Maximizing area contained by a fence.

Answer.
\(4410\ {\rm ft^{2}}\)

4.4.5.4. Minimizing the area of a poster.

Answer 1.
\(15.3137\ {\rm cm}\)
Answer 2.
\(45.9411\ {\rm cm}\)

4.4.5.5. Maximizing the area of a rectangle.

Answer 1.
\(3.83\)
Answer 2.
\(7.33333\)

4.4.5.6. Maximizing the volume of a closed box.

Answer.
The absolute maximum volume is \(V\left( \sqrt{\frac{5}{3}} \right) = \frac{15}{12}\left( \sqrt{\frac{5}{3}} \right) - \frac{1}{4}\left( \sqrt{\frac{5}{3}} \right)^3 \approx 1.07583\) cubic feet.

4.4.5.7. Maximizing pasture area with limited fencing.

Answer.
Exercise Answer

4.4.5.8. Minimizing cable length.

Answer.
\(172.047\) feet of cable.

4.4.5.9. Minimizing construction costs.

Answer.
The minimum cost is $1165.70.

4.5 Tangent Line Approximations
4.5.6 Exercises

4.5.6.1. Approximating value of unknown function.

Answer 1.
\(3\)
Answer 2.
\(3\)
Answer 3.
\(8\)
Answer 4.
\(2.2\)

4.5.6.2. Approximating value of square root.

Answer 1.
\(0.142857x+7\)
Answer 2.
\(7.02857\)
Answer 3.
\(\sqrt{x^{2}+2x+49}-0.142857x-7\)
Answer 4.
\(0.00278693\)

4.5.6.3. Approximating value of natural log.

Answer 1.
\(-5x+0\)
Answer 2.
\(0.4\)
Answer 3.
\(\log\!\left(-5x+1\right)-\left(-5\right)x-0\)
Answer 4.
\(-0.0635278\)

4.5.6.6. Using marginal profit.

Answer 1.
\(1155+12\)
Answer 2.
\(1155+4\cdot 12\)
Answer 3.
\(1155-12\)

4.6 Elasticity of Demand
4.6.5 Exercises

4.6.5.2. Maximizing revenue.

Answer 1.
\(\frac{450p^{2}}{225\!\left(150-p^{2}\right)}\)
Answer 2.
\(\sqrt{\frac{150}{3}}\)

4.6.5.3. Interpreting elasticity and revenue.

Answer 1.
\(0.01p\)
Answer 2.
\(0.07\)
Answer 3.
\(\frac{1}{0.01}\)

4.7 Derivatives of Functions Given Implicitly
4.7.3 Exercises

4.7.3.1. Implicit differentiation in a polynomial equation.

Answer.
\(\frac{1-5x^{4}y}{x^{5}-3}\)

4.7.3.2. Implicit differentiation in an equation with logarithms.

Answer.
\(\frac{y\!\left(8-x\ln\!\left(y\right)\right)}{x\!\left(x+2y^{2}\right)}\)

4.7.3.3. Slope of the tangent line to an implicit curve.

Answer.
\(-{\frac{7}{11}}\)

4.7.3.4. Equation of the tangent line to an implicit curve.

Answer.
\(x+7y = 10\)

4.8 Related Rates
4.8.3 Exercises

4.8.3.1. Related rates with polynomial equations.

Answer.
\(\frac{-2\cdot 4\cdot 5^{3}+2\cdot \left(-2\right)\!\left(-3\right)^{4}}{2\cdot 5\cdot 4\cdot \left(-3\right)^{3}}\)

4.8.3.2. Height of a conical pile of gravel.

Answer.
\(\frac{120}{289\pi }\)

4.8.3.3. Movement of a shadow.

Answer.
\(6.4\)

4.8.3.4. Docking a boat.

Answer.
The boat is approaching the dock at a rate of \(\frac{13}{6} \approx 2.167\) feet per second.

4.8.3.5. A leaking conical tank.

Answer.
\(704196.4446\)

4.8.3.6. Filling a swimming pool.

Answer.
The depth of the water is increasing at
\begin{equation*} \left. \frac{dh}{dt}\right|_{h = 5} = 1.28 \end{equation*}
feet per minute. The depth of the water is increasing at a decreasing rate.

5 Integration
5.1 Antiderivatives from Formulas
5.1.4 Exercises

5.1.4.1. Finding Antiderivatives.

Answer.
\(5t^{\frac{1}{5}}\)

5.1.4.2. Finding Antiderivatives (Constants).

Answer 1.
\(x\)
Answer 2.
\(y\)
Answer 3.
\(\pi z\)

5.1.4.3. Finding Antiderivatives (Polynomials).

Answer 1.
\(\frac{x^{2}}{2}\)
Answer 2.
\(ax\)
Answer 3.
\(t^{2}x\)

5.1.4.4. Finding Antiderivatives (Exponential).

Answer.
\(\frac{1}{3}e^{3x}\)

5.1.4.5. Finding Antiderivatives.

Answer.
\(\frac{2}{5}t^{\frac{5}{2}}-\frac{2}{1t^{\frac{1}{2}}}+C\)

5.1.4.6. Finding Antiderivatives (Simplifying).

Answer.
\(-0.666667x^{3}+1.5x^{2}+\left(-1\right)x+\left(-6\right)\ln\!\left(x\right)+C\)

5.1.4.8. Applying Antiderivatives.

Answer 1.
\(8t-2t^{2}+72\)
Answer 2.
\(8\cdot 5-2\cdot 5^{2}+72\)

5.2 Determining Area Under a Curve
5.2.9 Exercises

5.2.9.1. Evaluating definite integrals from graphical information.

Answer 1.
\(-21\)
Answer 2.
\(7\)
Answer 3.
\(-14\)
Answer 4.
\(28\)

5.2.9.2. Estimating definite integrals from a graph.

Answer 1.
\(-4\)
Answer 2.
\(-A\)

5.2.9.3. Riemann sum estimates and definite integrals.

Answer.
  1. The total change in position, \(P\text{,}\) is \(P = \int_0^1 v(t) \, dt + \int_1^3 v(t) \, dt + \int_3^4 v(t) \, dt = \int_0^4 v(t) \, dt\text{.}\)
  2. \(P = \int_0^4 v(t) \, dt \approx 2.665\text{.}\)
  3. The total distance traveled, \(D\text{,}\) is \(D = \int_0^1 v(t) \, dt - \int_1^3 v(t) \, dt + \int_3^4 v(t) \, dt\text{.}\)
  4. \(D \approx 8.00016\text{.}\)
  5. \begin{equation*} v_{\operatorname{AVG} [0,4]} \approx = 0.66625 \end{equation*}
    feet per second.

5.2.9.4. Using definite integrals on a velocity function.

Answer.
  1. The total change in position is \(P = \int_0^{4} v(t) dt\text{.}\)
  2. \(P = -2.625\) feet.
  3. \(D = 3.375\) feet.
  4. \(AV = -0.65625\) feet per second.
  5. \(s(t) = -t^2+t\text{.}\)

5.3 The Definite Integral
5.3.4 Exercises

5.3.4.1. Evaluating definite integrals.

Answer 1.
\(42.6666666666667\)
Answer 2.
\(6\)
Answer 3.
\(48.75\)

5.3.4.3. Area under a curve with no bounds.

Answer.
\(4.89897948556636\)

5.3.4.4. Application of definite integrals.

Answer.
\(3\cdot \left(9500-2000\right)-0.005\cdot \left(9500^{2}-2000^{2}\right)+2\times 10^{-6}\!\left(9500^{3}-2000^{3}\right)\)

5.3.4.5. Another application of definite integrals.

Answer.
\(11520\)

5.4 Properties of Integration
5.4.5 Exercises

5.4.5.1. Finding the average value of a linear function.

Answer.
\(29.5\)

5.4.5.2. Finding the average value of a function given graphically.

Answer 1.
\(\frac{1}{4}\)
Answer 2.
\(\frac{3}{8}\)
Answer 3.
\(0\)

5.4.5.3. Estimating a definite integral and average value from a graph.

Answer.
\(5\)

5.4.5.4. Using rules to combine known integral values.

Answer 1.
\(2-9-6\)
Answer 2.
\(-2\cdot \left(-13\right)+9\cdot 13-9\cdot 10\)

5.4.5.6. Area between two curves.

Answer.
\(\frac{2}{3}+2\cdot 2-\frac{2\cdot 3}{3}\)

5.4.5.7. Using the Sum and Constant Multiple Rules.

Answer.
  1. \(\int_0^1 [f(x) + g(x)] \,dx = 1-\frac{\pi}{4}\text{.}\)
  2. \(\int_1^4 [2f(x) - 3g(x)] \, dx = -\frac{15}{2} - 3\pi\text{.}\)
  3. \(h_{\operatorname{AVG} [0,4]} = \frac{5}{8} + \frac{3\pi}{16}\text{.}\)
  4. \(c = -\frac{3}{8} + \frac{3\pi}{16}\text{.}\)

5.4.5.8. Finding the area of a bounded region.

Answer.
  1. \(A_1 = \int_{-1}^{1} (3-x^2) \, dx\text{.}\)
  2. \(A_2 = \int_{-1}^{1} 2x^2 \, dx\text{.}\)
  3. The exact area between the two curves is \(\int_{-1}^{1} (3-x^2) \, dx - \int_{-1}^{1} 2x^2 \, dx\text{.}\)
  4. Use the sum rule for definite integrals over the same interval.
  5. Think about subtracting the area under \(q\) from the area under \(p\text{.}\)

5.6 Integration by Substitution
5.6.5 Exercises

5.6.5.1. Product involving a 4th power polynomial.

Answer.
\(\frac{\left(t^{4}-6\right)^{4}}{16}+C\)

5.6.5.2. Fraction involving \(\ln^9(x)\).

Answer.
\(0.166667\ln^{6}\!\left(z\right)+C\)

5.6.5.3. Fraction involving \(e^{5 x}\).

Answer.
\(0.25\ln\!\left(e^{4x}+7\right)+C\)

5.6.5.4. Fraction involving \(e^{5 \sqrt{y}}\).

Answer.
\(\frac{12e^{3\sqrt{y}}}{3}+C\)

5.6.5.5. Working with negative exponents.

Answer.
\(\frac{\left(x+13\right)^{-1}}{-1}\)

5.6.5.6. Fraction involving sums of exponential functions.

Answer.
\(\ln\!\left(\left|10e^{x}-7e^{-x}\right|\right)\)

5.6.5.7. Integral involving a rational function.

Answer.
\(\frac{0.2\!\left(x^{5}+\left(-9\right)\right)^{-4}}{-4}+C\)

5.6.5.8. Integral of a partial fraction.

Answer.
\(1.02961941718116\)

5.6.5.9. Find the value of a definite integral based on another.

Answer.
\(-2.5\)

5.6.5.10. A clever substitution.

Answer.
  1. \(\int x \sqrt{x-1} \, dx = \int (u+1) \sqrt{u} \, du\text{.}\)
  2. \(\int x \sqrt{x-1} \, dx = \frac{2}{5} (x-1)^{\frac{5}{2}} + \frac{2}{3} (x-1)^{\frac{3}{2}} + C\text{.}\)
  3. \(\int x^2 \sqrt{x-1} \, dx = \frac{2}{7} (x-1)^{\frac{7}{2}} + \frac{4}{5} (x-1)^{\frac{5}{2}} + \frac{2}{3} (x-1)^{\frac{3}{2}} + C\text{.}\)
    \(\int x \sqrt{x^2 - 1} \, dx = \frac{1}{3} (x^2-1)^{\frac{3}{2}} + C\text{.}\)

5.6.5.11. Definite integral with a clever substitution.

Answer.
\(\frac{-378-128}{15}\)

5.6.5.12. Integral involving a square root of a linear expression.

Answer.
\(\frac{-8}{3}\!\left(4-x\right)\sqrt{4-x}+\frac{2}{5}\!\left(4-x\right)^{2}\sqrt{4-x}\)
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