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
\(ln(x+2)=12\)
\(e^{3x+1}=16\)
\(6e^{4x}=41e^{2x}\)
2) Find the derivative:
\[ G(a)=\frac{2\ln(4a-2)}{e^{3a}} \]
Reimann Sum
- 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? Desribe in words,show mathmatically, or draw on the graph how evaluating the rectangle in different ways might lead to slighly different approximations.
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 \(Q^*\))
2)
Take the Integrals
\[\begin{align} \text{A) }y=\frac{3}{x^2}, y(0)=5& &\text{B) }g(t)=3t^5-2t^3+16t-7 & &\text{C) } \int^4_2\frac{1}{2}x \end{align}\]