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Appendix A Answers to Selected Exercises

Below you will find answers to selected exercises from the end of each section. Worked solutions are not provided, so you are encouraged to discuss your thought process and reasoning for exercises with others, even if your final answer matches what is listed here. Your process and reasoning are the most important things for deepening your understanding and ensuring you can use important concepts on your own in the future, so use this section as a guide, but not a replacement for engaging in the exercises with others.

1 Functions As Models
1.1 Biology and Calculus
1.1.4 Exercises

1.2 Functions
1.2.8 Exercises

1.3 Units and Dimensions of Functions
1.3.5 Exercises

1.4 Linear Functions
1.4.5 Exercises

1.4.5.2.

Answer.
  1. \(\displaystyle y-10 = \frac{-13}{6}(x+2)\)
  2. \(\displaystyle y = \frac{-13}{6}x + \frac{17}{3}\)

1.4.5.3.

Answer.
\(x = \frac{-4}{m-2}\text{,}\)\(m \neq 2\)

1.5 Exponential and Logarithmic Functions
1.5.5 Exercises

1.5.5.4.

Answer.
\(\log(b)\text{,}\)\(y\)\((0,\log(a))\)

1.6 Trigonometric Functions
1.6.4 Exercises

1.6.4.3.

Answer.
  1. \(c(t)=2\cos\left(\frac{\pi}{12}t\right) + 3\text{,}\) \(u(t)=0.5\cos\left(\frac{\pi}{2}t\right) + 1\)

1.6.4.4.

Answer.
\(x = \frac{\pi}{6} + 2\pi k\text{,}\) \(x = \frac{5\pi}{6} + 2\pi k\text{,}\) where \(k\) is any integer
\(2\) solutions in \([0,\pi]\)
\(0\) solutions in \([-\pi,0]\)

1.7 Discrete-Time Dynamical Systems
1.7.4 Exercises

1.7.4.2.

Answer.
  1. \(\displaystyle b_t = t+5\)
  2. \(\displaystyle b_t = 5\cdot (0.75)^t\)

1.7.4.3.

Answer.
  1. \(b_{t+1} = 0.4b_t + 3\text{,}\) \(b_0 = 10 \)
  3. \(\displaystyle b_t = 5\cdot (0.4)^t + 5\)

1.8 Analyzing Discrete-Time Dynamical Systems
1.8.4 Exercises

1.9 Applications: The Lung Model and Competing Species
1.9.4 Exercises

1.9.4.1.

Answer.
  2. \(\displaystyle c^* = \dfrac{p\beta}{p + \alpha(1-p)}\)
  3. Table 1.9.13.
    \(\alpha\) \(c^*\)
    \(0\) \(0.2\)
    \(0.25\) \(0.16\)
    \(0.5\) \(0.133\)
    \(0.75\) \(0.114\)
    \(1\) \(0.1\)

1.9.4.4.

Answer.
  2. \(x_{t+1} = 1.6x_t + 0.3y_t\text{,}\) \(y_{t+1} = 0.4x_t + 0.7y_t\)
  3. \(\displaystyle p_{t+1} = \dfrac{1.3p_t + 0.3}{p_t + 1}\)
  4. \(p^* = 0.72\text{,}\) stable

1.9.4.5.

Answer.
  1. \(\displaystyle k_{t+1} = 0.2k_t +0.75h_t\)
  2. \(\displaystyle h_{t+1} = 0.8k_t +0.25h_t\)
  3. \(\displaystyle p_{t+1} = 0.75 - 0.55p_t\)
  4. \(p^*= 0.484\text{,}\) stable

2 The Derivative
2.1 Limits of Functions
2.1.4 Exercises

2.1.4.4.

Answer.
  1. \(\displaystyle AROC_{[1,1+h]} = \dfrac{100\cos(0.75(1+h)) \cdot e^{-0.2(1+h)} - 100\cos(0.75)\cdot e^{-0.2}}{h}\)
  2. \(\displaystyle -53.8\)
  3. feet/s

2.2 The Derivative of a Function at a Point
2.2.3 Exercises

2.2.3.1.

Answer.
  2. \(AROC_{[-3,-1]} \approx \frac{2.3}{2} = 1.15\)
    \(AROC_{[0,2]} \approx \frac{-0.8}{2} = -0.4\)
  3. \(IROC_{x=-3} \approx \frac{5}{2} = 2.5\)
    \(IROC_{x=0} \approx \frac{-1}{3} \)

2.2.3.3.

Answer.
  1. \(P(7) - P(0) \approx .118\) billion people
    \(AROC_{[0,7]} \approx 0.017\) billion people per year
    \(AROC_{[0,7]} \lt IROC_{t=7}\)
  2. \(AROC_{[19,29]} \approx 0.0223\) billion people per year
  3. If today is July 1, 2022, then the limit would be \(\displaystyle \lim_{h \to 0} \frac{P(29.5 +h) - P(29.5)}{h} \approx 0.024\) billion people per year
  4. If today is July 1, 2022, the equation of the tangent line would be \(y= 1.733 +0.024(t - 29.5)\)

2.2.3.4.

Answer.
  1. \(\displaystyle f'(2) = 1\)
  2. \(\displaystyle f'(1) = -1\)
  3. \(\displaystyle f'(1) = 0.5\)
  4. \(f'(1)\) does not exist
  5. \(\displaystyle f'\left(\frac{\pi}{2}\right) = 0\)

2.3 The Derivative Function
2.3.3 Exercises

2.3.3.3.

Answer.
  1. \(g\) is linear with slope \(1\) on \((0,2)\)
  2. \((-\infty,-2)\text{,}\) \((-2,0)\text{,}\)\((2,\infty)\)
  3. \(\displaystyle x = -2, 0, 2\)

2.4 The Second Derivative
2.4.5 Exercises

2.4.5.1.

Answer.
  1. \(f\) is increasing, concave down near \(x=2\)
  2. It is likely that \(f(2.1) \gt -3\)
  3. It is likely that \(f'(2.1) \lt 1.5\)

2.4.5.2.

Answer.
  1. \(\displaystyle g'(2) \approx 1.4\)
  2. \(g\) can have at most one real zero
  3. \(\displaystyle 9\)
  4. \(\displaystyle g''(2) \approx 5\)

2.5 Elementary Derivative Rules
2.5.5 Exercises

2.5.5.1.

Answer.
  1. \(h(2) = 27\text{,}\) \(h'(2) =\frac{-19}{2}\)
  2. \(\displaystyle y= 27 - \frac{19}{2}(x-2)\)
  3. Increasing

2.5.5.2.

Answer.
  1. \(p'\) and \(q'\) both do not exist when \(x= \pm 1\)
  2. \(r'(-2)=4\text{,}\) \(r'(0)=0.5\)
  3. \(\displaystyle y=4\)

2.6 Derivatives of the Sine and Cosine Functions
2.6.3 Exercises

2.6.3.1.

Answer.
  1. \(V'(2)= -0.638\) thousand dollars per year
  2. \(V''(2) = -5.33\) thousand dollars per year per year

2.6.3.2.

Answer.
  1. \(\displaystyle f'\left(\frac{\pi}{4}\right) = \frac{-5\sqrt{2}}{2}\)
  2. \(\displaystyle y = 3 + 2(x-\pi)\)
  3. Decreasing
  4. Above the curve

2.6.3.3.

Answer.
    1. \(\displaystyle s(\theta) = \cos(\theta)\)
    2. \(\displaystyle s'(\theta) = -\sin(\theta)\)
    3. \(\displaystyle s'(\theta) = \cos\left(\theta +\frac{\pi}{2}\right)\)

2.7 Derivatives of Products and Quotients
2.7.5 Exercises

2.7.5.2.

Answer.
  1. \(h(2) = -15\text{,}\) \(h'(2) = \frac{23}{2}\)
  2. \(\displaystyle y=-15 + \frac{23}{2}(x-2)\)
  3. Increasing
  4. \(\displaystyle y=\frac{-3}{5} + \frac{17}{50}(x-2)\)

2.7.5.3.

Answer.
  1. \(r'(-2) = 5\text{,}\) \(r'(0) = 1\)
  2. \(\displaystyle y=2\)
  3. \(z'(0)=-4\text{,}\) \(z'(2)=-1\)
  4. \(\displaystyle \ell'(0) = 0\)

2.7.5.4.

Answer.
  1. \(\displaystyle C(t) = A(t) \cdot Y(t)\)
  2. \(C(0) = 1,190,000\) bushels
  3. \(\displaystyle C'(t) = A'(t)Y(t) + A(t)Y'(t)\)
  4. \(C'(0) = 158,000\) bushels per year
  5. \(\displaystyle y=1,190,000 + 158,000t\)

2.7.5.5.

Answer.
  1. \(g(v) = \frac{1}{f(v)}\text{,}\) \(g(80)= 20\) km/L, \(g'(v) = -0.16\) (km/L)/(km/h)
  2. \(h(v)=f(v) \cdot v\text{,}\) \(h(80)= 4\) L/h, \(h'(80) = 0.082\) (L/h)/(km/h)

2.8 Derivatives of Compositions
2.8.5 Exercises

2.8.5.1.

Answer.
  1. \(\displaystyle h'\left(\frac{\pi}{4}\right) = \frac{3\sqrt{2}}{4}\)
  2. \(r\) is changing most rapidly

2.8.5.2.

Answer.
  1. \(\displaystyle p'(x) = e^{u(x)}\cdot u'(x)\)
  2. \(\displaystyle q'(x) = u'(e^x)\cdot e^x\)
  3. \(\displaystyle r'(x) = -\sin(u(x))\cdot u'(x)\)
  4. \(\displaystyle s'(x) = u'(\cos(x))\cdot (-\sin(x))\)
  5. \(\displaystyle a'(x) = u'(x^4)\cdot 4x^3\)
  6. \(\displaystyle b'(x) = 4(u(x))^3\cdot u'(x)\)

2.8.5.3.

Answer.
  1. \(C'(0)=0\text{,}\) \(C'(3) = -0.5\)
  2. \(Y'(3) = 0\text{,}\) \(Z'(0) = 0\)

2.8.5.4.

Answer.
  1. \(7 \pi\) cubic feet per foot
  2. \(h'(2) = \pi\) feet per hour
  3. \(7\pi^2\) cubic feet per hour

2.9 Derivatives of Inverse Functions
2.9.5 Exercises

2.9.5.3.

Answer.
  2. \(\displaystyle g(x) = (4x-16)^{\frac{1}{3}}\)
  3. \(f'(2)=3\text{,}\) \(g'(6) = \frac{1}{3}\)

3 Using the Derivative
3.1 Linear and Quadratic Approximation
3.1.5 Exercises

3.1.5.2.

Answer.
  1. \(s(9.34) \approx 3.592\) feet
  2. Underestimate
  3. \(Q(9.34)= 3.596624\) feet
  4. Moving towards the origin, slowing down

3.1.5.3.

Answer.
  1. \(\displaystyle x = 1\)
  2. \(\displaystyle (-0.35, 1.3)\)
  3. \(f(1.88) \approx -3.0022\text{,}\) overestimate
  4. \(\displaystyle Q(x) = -3 + e^{-4}(x-2) - \frac{3}{2}e^{-4}(x-2)^2\)

3.2 The Stability Theorem
3.2.4 Exercises

3.2.4.3.

Answer.
  1. \(y^*=0\) (unstable), \(y^*=1,000\) (stable)
  2. \(b^*=1\) (stable), \(b^*=2\) (unstable)

3.3 The Logistic Discrete-Time Dynamical System
3.3.4 Exercises

3.3.4.1.

Answer.
  1. \(x^*=0\) (unstable), \(x^*=0.5\) (stable)
  3. The solution looks to increase exponentially at first (concave up), but then there is an inflection point where the solution continues to increase but is concave down
  4. \(x^*=0.5\) is a horizontal asymptote in the solution function graph

3.4 Identifying Extreme Values of Functions
3.4.4 Exercises

3.4.4.1.

Answer.
  1. Critical numbers are \(x=-1\) (local min) and \(x=1\) (neither)
  3. \(f\) is CCU on \((-\infty, -0.3) \cup (1,\infty)\) and CCD on \((-0.3,1)\text{.}\) \(f\) has inflection points at \(x = -0.3, 1\text{.}\)

3.4.4.2.

Answer.
  1. Neither
  2. \(g''\) will change from negative to positive at \(x=2\)
  3. It is an inflection point

3.4.4.3.

Answer.
  1. Inflection points at \(x=-1, 2\)
  2. Local maximum
  3. \(y = \frac{12}{e^2} - \frac{5}{e^2}(x-2)\text{,}\) neither above nor below the curve

3.5 Global Optimization and Applications
3.5.4 Exercises

3.5.4.2.

Answer.
  1. Max of \(y=30\) at \(x=1\text{,}\) Min of \(y=24\) at \(x=2\)
  2. Max of \(y=40\) at \(x=6\text{,}\) Min of \(y=24\) at \(x=2\)
  3. Max of \(y=40\) at \(x=6\text{,}\) Min of \(y=26\) at \(x=3\)

3.6 Limits: L’Hôpital’s Rule
3.6.4 Exercises

3.6.4.3.

Answer.
  1. \(\displaystyle x^3\)
  2. \(\displaystyle 5^x\)
  3. \(\displaystyle \sqrt{x}\)
  4. \(\displaystyle \sqrt[n]{x}\)

3.7 Limits: Leading Behaviors
3.7.4 Exercises

3.7.4.1.

Answer.
  1. \(\displaystyle k_\infty(x)= e^x\)
  2. \(\displaystyle \displaystyle \lim_{x \to \infty}\frac{e^x}{e^x + e^{-x}} = 1\)
  3. \(\displaystyle k_{-\infty}(x) = e^{-x}\)
  4. \(\displaystyle \displaystyle \lim_{x \to -\infty}\frac{e^x}{e^x + e^{-x}} = 0\)

4 Continuous-Time Dynamical Systems
4.1 Introduction to Differential Equations and Antiderivatives
4.1.4 Exercises

4.1.4.1.

Answer.
\(q(t) = \ln(t) + 0.25t^4 - \frac{2^t}{\ln(2)} + C \)

4.1.4.2.

Answer.
\(p(t) = \sin(t) + \frac{2}{5}t^{\frac{5}{2}} +1 \)

4.2 Solving Pure-Time Differential Equations
4.2.4 Exercises

4.2.4.1.

Answer.
\(s(t)= -\sin(t) + \frac{1}{6}t^3 -t +5\)

4.2.4.2.

Answer.
  1. \(\displaystyle 0.5(x^4 + 4x^2 + C)\)
  2. \(\displaystyle (\ln(x))^2 + 4\ln(x) +C\)

4.2.4.4.

Answer.
  2. \(A(t) = 4t - \frac{1}{0.263} \cos(0.263t +4.7) + \frac{1}{0.526}\sin(0.526t+9.4) + C\text{,}\) measured in pounds

4.3 Riemann Sums
4.3.6 Exercises

4.3.6.1.

Answer.
  1. \(\Delta x = 0.75\text{,}\) \(L_4=40.125\text{,}\) \(R_4 = 46.875\)
  2. \(L_4\) is an underestimate and \(R_4\) is an overestimate of the actual area of \(43.5\)

4.3.6.2.

Answer.
  1. \(f(x)=x^2+1\) on \([1,3]\)
  2. Left: \(f(x) =x^2 + 1\) on \([1.4,3.4]\)
    Middle: \(f(x) =x^2 + 1\) on \([1.2,3.2]\)
  3. The area under the graph of \(x^2+1\) from \(x=1\) to \(x=3\)
  4. \(n=10\text{,}\) \(\Delta x = 0.2\text{,}\) \(R_{10}= \sum_{i=1}^{10} [(1+0.2i)^2 + 1] \cdot 0.2\)

4.3.6.3.

Answer.
  2. \(M_3 = 99.6\) feet
  3. \(L_6 = 114\) feet, \(R_6 = 84\) feet, \(\frac{1}{2}(L_6 + R_6) = 99\) feet
  4. \(\displaystyle L_6\)

4.3.6.4.

Answer.
  1. \(\displaystyle M_4 \approx 6.45\)
  2. Units of \(M_4\) are tons
  3. \(\displaystyle L_5 = 5.196\)
  4. \(\displaystyle R_4 = 8.119\)

4.4 The Definite Integral
4.4.5 Exercises

4.4.5.1.

Answer.
  1. \(\displaystyle \int_0^4 v(t) dt\)
  2. \(\frac{-21}{8}\) feet
  3. \(\int_0^{0.5} v(t) dt - \int_{0.5}^{3.5} v(t) dt + \int_{3.5}^4 v(t) dt = \frac{27}{8}\) feet
  4. \(\frac{-21}{32}\) feet per second
  5. \(\displaystyle s(t) = -t^2 + t\)

4.4.5.2.

Answer.
  1. \(\displaystyle \int_0^4 t(t-1)(t-3) dt\)
  2. \(2.66\) feet
  3. \(\displaystyle \int_0^1 v(t) dt - \int_1^3 v(t)dt + \int_3^4 v(t)dt\)
  4. \(8.00\) feet
  5. \(0.665\) feet per second

4.4.5.3.

Answer.
  1. \(\displaystyle 1 - \frac{\pi}{4}\)
  2. \(\displaystyle \frac{-15}{2}-3\pi\)
  3. \(\displaystyle \frac{5}{8} + \frac{3\pi}{16}\)
  4. \(\displaystyle c= \frac{-3}{8} + \frac{3\pi}{16}\)

4.4.5.4.

Answer.
  1. \(\displaystyle \int_{-1}^1 3 - x^2 dx\)
  2. \(\displaystyle \int_{-1}^1 2x^2 dx\)
  3. \(\displaystyle \int_{-1}^1 3 - x^2 dx - \int_{-1}^1 2x^2 dx \)

4.5 The Fundamental Theorem of Calculus
4.5.5 Exercises

4.5.5.2.

Answer.
  1. \(m(h) = \frac{1}{c(h)}\text{,}\) measured in minutes per foot
  2. Input: height in feet, Output: number of minutes
  3. \(\displaystyle \int_0^{10,000} m(h) dh\)
  4. \(M_5 = 15.27\) minutes

4.6 Approximations of Solutions
4.6.4 Exercises

4.6.4.1.

Answer.
  1. Alice: \(\frac{dT}{dt} = -15\) degrees F per minute
    Bob: \(\frac{dT}{dt} = -3\) degrees F per minute
  2. Bob’s cup is better insulated

4.6.4.2.

Answer.
  2. \(\displaystyle \frac{d^2y}{dt^2} = 1 - \frac{dy}{dt}\)
  3. \(y(1) \approx 0.7414796816\) using quadratic approximations, which is closer to the actual value of \(y(1) = 0.735758882\) than using Euler’s method
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