Calculus Workshop

Integration Problem Set

Answer the following questions to the best of your ability. Feel free to work with anyone in the cohort, though I would encourage attempting on your own first to make sure you fully understand the concepts.

Logarithms

1) Solve for x

\(ln(x+2)=12\)

\[\begin{align} x+2=e^{12}\\ x=e^{12}-2\\ x=162,752.79 \end{align}\]

\(e^{3x+1}=16\)

\[\begin{align} 3x+1=\ln(16)\\ x=\frac{\ln(16)-1}{3} \\ x=1.26 \end{align}\]

\(6e^{4x}=41e^{2x}\)

\[\begin{align} \frac{e^{4x}}{e^{2x}=\frac{41}{6}}\\ e^{4x-2x}=\frac{41}{6} \\ e^{2x}=\frac{41}{6}\\ 2x=\ln(41)-\ln(6)\\ x=\frac{\ln(41)-\ln(6)}{2} \end{align}\]

2) Find the derivative:

\[ G(a)=\frac{2\ln(4a-2)}{e^{3a}} \]

\[\begin{align} \frac{dG}{da}=\frac{\frac{8e^{3a}}{4a-2}-3e^{3a}\ln(4a-2)}{[e^{3a}]^2} \end{align}\]

Reimann Sum

1. Approximate the area under the curve \(f(x)=3x^2-6x+10\) on the interval \([6,12]\) with 6 uniform rectangles.

Does the position of the rectangle make a difference? Describe in words,show mathematically, or draw on the graph how evaluating the rectangle in different ways might lead to slightly different approximations.

x=seq(6,12)

f=3*x^3-6*x+10

sum(f)

[1] 17269

Intergrals

1)

The marginal benefit of abatement (e.g. reducing) for carbon is given by:

\[ MB=31-2Q \]

However there is also a marginal cost for carbon abatement given by:

\[ MC=6+3Q \]

Find the total net benefit of carbon abatement to society at equilibrium. (Hint: To get equilibrium and the bounds of the integral, first set marginal benefit equal to marginal cost and solve for \(Q^*\). Then your integral bounds should be from 0 to \(Q^*\))

\[\begin{align} 31-2Q=6+3Q\\ 25=5Q \\ 5=Q^* \end{align}\]

\[\begin{align} \int^5_030-2Qdq \\ 30Q-Q^2|^5_0 \\ 30(5)-(5)^2=125 \end{align}\]

\[\begin{align} \int^5_06+3Q dq\\ 6Q+\frac{3x^2}{2}|^5_0\\ 6(5)+\frac{3(5)^2}{2}=67.5 \end{align}\]

2)

Take the Integrals

\[\begin{align} \text{A) }y=\frac{3}{x^2}, y(1)=5& &\text{B) }g(t)=3t^5-2t^3+16t-7 & &\text{C) } \int^4_2\frac{1}{2}x \end{align}\]

1. \[\begin{align} \int\frac{3}{x^2} dx\\ -3x^{-3}+C\\ -3(1)^{-3}+C=5\\ C=8\\ x^{-3}+8 \end{align}\]

2. \[\begin{align} \int\ g(t)=3t^5-2t^3+16t-7 dt\\ \frac{t^6}{2}-\frac{t^4}{2}+8t^2-7t+C\\ \end{align}\]

3. \[\begin{align} \int^4_2\frac{1}{2}x dx\\ \frac{1}{4}x^2|^4_2\\ \frac{1}{4}(4)^2-\frac{1}{4}(2)^2=3 \end{align}\]