The problems are organized by lecture. Students should show derivations, state assumptions, and interpret results. Selected guidance is included to clarify the intended mathematical method; it is not a substitute for complete proofs.
Set 1 — Limits and continuity
- Prove \(\lim_{x\to2}(3x+1)=7\) using epsilon-delta.
- Evaluate \(\lim_{x\to0}(\sqrt{1+x}-1)/x\).
- Construct examples of removable, jump, and infinite discontinuities.
Guidance: For Question 1, choose \(\delta=\varepsilon/3\), since \(|3x+1-7|=3|x-2|\). For Question 2, rationalize the numerator before taking the limit.
Set 2 — Sequences and series
- Determine convergence of \(a_n=(2n+1)/(n+3)\).
- Test \(\sum_{n=1}^\infty1/[n(n+1)]\) and find its sum.
- Determine absolute or conditional convergence of \(\sum(-1)^{n+1}/n\).
Guidance: Use the sequence limit for Question 1. For Question 2, use partial fractions and telescoping. For Question 3, apply the alternating-series test and separately test absolute convergence.
Set 3 — Differentiation
- Differentiate \(x^2e^{3x}\).
- Differentiate \(\log(1+x^2)\).
- Use the mean value theorem to prove a monotonicity result.
Guidance: Apply product and chain rules; state the interval and regularity assumptions for the theorem.
Set 4 — Taylor and optimization
- Obtain the second-order Taylor polynomial of \(e^x\) at zero.
- Find and classify stationary points of \(f(x)=x^3-3x\).
- Maximize \(20q-q^2-16\) over \(q\ge0\).
Guidance: Solve the FOC, apply the SOC, and compare with boundary values when the domain is constrained.
Set 5 — Integration
- Evaluate \(\int_0^1xe^{x^2}dx\).
- Evaluate \(\int xe^x dx\).
- Determine when \(\int_1^\infty x^{-p}dx\) converges.
Guidance: Use substitution, integration by parts, and an improper-integral limit, respectively.
Set 6 — Multivariable limits
- Test the limit of \(xy/(x^2+y^2)\) at the origin.
- Describe level curves of \(x^2+4y^2=c\).
- Explain the role of compactness in the extreme value theorem.
Guidance: Two paths with distinct limiting values disprove a multivariate limit.
Set 7 — Partial derivatives and chain rule
- Compute the gradient and Hessian of \(f(x,y)=x^2+xy+2y^2\).
- Derive marginal products for Cobb–Douglas production.
- Apply the multivariate chain rule to a composite function.
Set 8 — Unconstrained optimization
- Classify stationary points using Hessian eigenvalues.
- Maximize a strictly concave quadratic.
- Explain why semidefinite curvature is inconclusive.
Set 9 — Lagrange multipliers
- Maximize \(x^\alpha y^{1-\alpha}\) subject to \(p_xx+p_yy=m\).
- Interpret the multiplier as the marginal value of income.
- Derive the expenditure-minimization dual problem.
Guidance: Log-transform the Cobb–Douglas objective, derive both FOCs, and substitute into the budget constraint.
Set 10 — KKT conditions
- Minimize \((x-3)^2\) subject to \(x\le2\).
- Identify active and inactive constraints in a quadratic program.
- State a sufficient convexity condition for global optimality.
Set 11 — Implicit functions
- Derive \(dy/dx\) from \(F(x,y)=0\).
- Compute equilibrium comparative statics in a supply-demand system.
- Explain why a nonsingular Jacobian gives local, not necessarily global, invertibility.
Set 12 — Multiple integrals
- Reverse the order of integration over a triangular region.
- Derive the polar-coordinate Jacobian.
- Obtain marginal densities and expectations from a joint density.
Set 13 — Differential equations
- Solve a separable first-order ODE.
- Solve a linear ODE using an integrating factor.
- Determine local stability from a Jacobian matrix.
Set 14 — Difference equations
- Solve \(x_{t+1}=a+bx_t\).
- Classify stability for positive and negative slopes.
- Analyze a cobweb adjustment process.
Computational assignment
Use Python or R to:
- approximate derivatives and integrals;
- plot functions, contours, gradients, and Hessian eigenvalues;
- solve constrained optimization problems;
- simulate ODE and difference-equation paths;
- compare numerical output with analytical results.
Submit source code, a short methods note, and reproducible output.