Section 09: Exam Preparation
Work individually for 5 minutes, then discuss with the class (5 minutes)
Find the antiderivative: \(\int (4x^3 - 2x + 5) \, dx\)
Rewrite \(\dfrac{3}{\sqrt{x}}\) using a negative exponent, then integrate.
Evaluate: \(\int_0^1 e^{2x} \, dx\)
A marginal cost function is \(MC(x) = 6x + 20\). If fixed costs are 500, find the total cost function \(C(x)\).
Antiderivative Refresh (Sections 06 & 01)
Work in pairs
Find the antiderivatives of each expression. Rewrite first using exponent rules where needed!
(a) \(3x^{-2}\)
(b) \(e^{5x}\)
(c) \(\dfrac{1}{x}\)
(d) \(4\sqrt{x}\) (rewrite as \(4x^{1/2}\))
(e) \(\dfrac{2}{x^3}\) (rewrite as \(2x^{-3}\))
Definite Integral Computation (Section 06)
Work in pairs
Evaluate each definite integral:
(a) \(\displaystyle\int_0^2 (x^2 - 1) \, dx\)
(b) \(\displaystyle\int_1^e \frac{3}{x} \, dx\)
(c) \(\displaystyle\int_0^1 2e^{3x} \, dx\)
(d) \(\displaystyle\int_{-1}^{1} (x^3 - x) \, dx\) (exploit symmetry!)
Continuous Compounding (Sections 08 & 06)
Work in pairs
An income stream pays \(f(t) = 1000\) euro/year continuously, and the discount rate is \(r = 0.05\).
The present value over \(T = 10\) years is:
\[PV = \int_0^{10} 1000 \cdot e^{-0.05t} \, dt\]
(a) Compute this integral.
(b) Interpret the result: what does the present value tell us?
Area Between Curves (Sections 06 & 02)
Work in pairs
Find the area enclosed by the curves:
\[f(x) = x^2 \qquad \text{and} \qquad g(x) = 2x + 3\]
(a) Find the intersection points (use the quadratic formula from Sec 02).
(b) Determine which function is on top in the enclosed region.
(c) Set up and compute the integral for the enclosed area.
Consumer & Producer Surplus (Sections 06 & 08)
Work in pairs
A market has demand \(D(q) = 100 - 2q^2\) and supply \(S(q) = 10 + q^2\).
(a) Find the equilibrium price and quantity.
(b) Calculate consumer surplus.
(c) Calculate producer surplus.
(d) Interpret the results.
Substitution Method (Section 06)
Work in pairs
Evaluate each integral using substitution:
(a) \(\displaystyle\int 2x \cdot e^{x^2} \, dx\)
(b) \(\displaystyle\int \frac{3x}{x^2+4} \, dx\)
(c) \(\displaystyle\int x(x^2+1)^4 \, dx\)
Average Cost Word Problem (Sections 06 & 08)
Work in pairs
A company’s daily production cost rate is \(C(t) = 200 + 50e^{-0.1t}\) euro/hour over an 8-hour shift (\(0 \leq t \leq 8\)).
(a) Find the total production cost over the 8-hour shift: \(\int_0^8 (200 + 50e^{-0.1t}) \, dt\)
(b) Find the average hourly cost: \(\bar{C} = \frac{1}{8}\int_0^8 C(t) \, dt\)
(c) At what time \(t\) does \(C(t)\) equal the average hourly cost?
Area Between Exponential and Linear (Sections 06 & 01)
Work in pairs
Consider the functions \(f(x) = e^x\) and \(g(x) = x + 1\) on the interval \([0, 1]\).
(a) Which function is on top on \([0, 1]\)?
(b) Find the area enclosed between \(f(x)\) and \(g(x)\) on \([0, 1]\).
Integration by Parts (Section 06)
Think individually (2 min), then work in groups of 3-4
Evaluate the following integrals using integration by parts:
(a) \(\displaystyle\int x \cdot e^{-2x} \, dx\)
(b) \(\displaystyle\int x^2 \cdot \ln(x) \, dx\)
Factory Cost Reconstruction (Sections 06 & 08)
Think individually (2 min), then work in groups of 3-4
A factory’s marginal cost function is:
\[MC(x) = 0.06x^2 - 3x + 80\]
and fixed costs are 5000 euro.
(a) Find the total cost function \(C(x)\).
(b) Find the average cost function \(\overline{C}(x)\).
(c) Find the production level that minimizes average cost.
(d) Show that at this level, marginal cost equals average cost.
Pollution Accumulation Model (Section 06)
Think individually (2 min), then work in groups of 3-4
A factory releases pollutants at a rate of:
\[P(t) = 50 \cdot e^{-0.1t} \text{ tons/year}\]
(a) How much pollution is released in the first 5 years?
(b) How much pollution is released in the long run (\(t \to \infty\))?
(c) After how many years has 90% of the total long-run pollution been released?
Long-Run Accumulation (Section 06)
Think individually (2 min), then work in groups of 3-4
A machine generates waste at a rate of \(W(t) = 80 \cdot e^{-0.25t}\) kg/year.
(a) Find the total waste produced in the first \(T\) years: \(Q(T) = \int_0^T 80e^{-0.25t} \, dt\)
(b) Compute \(Q(10)\), \(Q(20)\), and \(Q(50)\).
(c) What value does \(Q(T)\) approach as \(T\) becomes very large?
(d) After how many years has 95% of the long-run total been produced? (Use logarithms.)
Break-Even Analysis via Integration (Sections 06 & 08)
Think individually (2 min), then work in groups of 3-4
A company launches a new product. The profit rate is \(P(t) = 6t - t^2 - 5 = -(t-1)(t-5)\) thousand euro/month, where \(t\) is months since launch.
(a) Which months is the company making a profit? (Find where \(P(t) > 0\).)
(b) Find the total profit over the profitable period.
(c) First month, the company operates at a loss. Total loss \(\int_0^1 P(t) \, dt\)?
(d) Find the net total outcome over the first 8 months: \(\int_0^8 P(t) \, dt\).
Total Profit from Marginal Analysis (Sections 06, 05 & 08)
Think individually (2 min), then work in groups of 3-4
A firm has:
(a) At what quantity \(x^*\) does \(MR = MC\)?
(b) Compute total profit: \(\displaystyle\int_0^{x^*} [MR(x) - MC(x)] \, dx\).
(c) Verify with \(R(0) = 0\) and \(C(0) = 500\).
Session 09-03 - Integral Calculus Review | Dr. Nikolai Heinrichs & Dr. Tobias Vlćek | Home