Math 243: Calculus II

Unit III Exam: Review Guide

1. Find the arc length of the curve \(y = \sqrt{2 - x^2}\), \(0 \leq x \leq 1\). Check your answer by using geometry.

2. Find the arc length function for the curve \(y = \arcsin(x) + \sqrt{1 - x^2}\) with starting point \((0,1)\).

3. Find the arc length function for the curve \(f(x) = \ln(\sin x)\), \(0 < x < \pi\), with starting point \(\left(\frac{\pi}{2},0\right)\).

4. Find the exact surface area of the surface obtained by rotating the curve about the \(y\)-axis. \[ y = \sqrt{1 + e^x}\, ,\ \ \ 0\leq x\leq 1 \]

5. Compute the volume and surface area of Gabriel's Horn: the region obtained by rotating the curve \(y = \dfrac{1}{x}\), \(x \geq 1\), about the \(x\)-axis.

6. Sketch the slope field for the differential equation, find the equilibrium solutions, and use the slope field to sketch a few integral curves. \[ y' = y^2 - 2y \]

7. Consider the initial value problem \[ \begin{cases} y' = ty - t^2, \\[1 ex] y(1) = 0 \end{cases} \] a.) Sketch the slope field at a few points, then use Euler's Method with a step size of \(h = 0.1\) to sketch the approximate solution curve. (Feel free to use a computer for this part. You need to know the ideas for the test, but you won't need to carry out the *entire* computation.)

b.) Find the exact solution and compare its graph to your approximation.

8. Find the general solution of the differential equation. \[ \dfrac{dL}{dt} = kL^2\ln(t) \]

9. Find the particular solution of the initial value problem. \[ \begin{cases} \dfrac{dP}{dt} = \sqrt{P\, t}, \\[1 ex] P(1) = 2. \end{cases}\]

10. Consider the integral equation \[ y(x) = 4 + \int_0^x 2t\sqrt{y(t)}\,dt. \] a.) Write the corresponding initial value problem.

b.) Find the solution of the integral equation by actually solving the initial value problem.

11. A sphere with radius 1m has temperature 15\(^\circ\)C. It lies inside a concentric sphere with radius 2m and temperature 25\(^\circ\)C. The temperature \(T = T(r)\) at a distance \(r\) from the common center of the spheres satisfies the second order differential equation \[ \dfrac{d^2 T}{dr^2} + \dfrac{2}{r} \dfrac{dT}{dr} = 0. \] Let \(S = T'(r)\) and rewrite the differential equation as a first order equation for \(S\). Solve for \(S\), then plug back in to solve for \(T\). Use the two boundary conditions for find the particular solution \(T = T(r)\).

12. A model for tumor growth is given by the Gompertz equation \[ \dfrac{dV}{dt} = a(\ln b - \ln V)V \] where \(a\) and \(b\) are positive constants and \(V = V(t)\) is the volume of the tumor measured in mm\(^3\). Find the general solution of the DE, then find the particular solution satisfying \(V(0) = 1\) mm\(^3\).

Teaching | Personal | Research

Back to main page

Your use of Wichita State University content and this material is subject to our Creative Common License.