Tutorials

Tutorial 1: Review of Algebra and Calculus

Outcomes: At the end of this tutorial you should be able to

- Understand the rules of the natural exponent and logarithmic functions.
- Solve equations containing these functions.
- Differentiate and integrate functions using the associated rules.
- Differentiate and integrate functions using the associated methods

Logarithmic Equations

Solve each of the following logarithmic equations for \(x\):

  1. \(\ln{(5x+4)}=0\)

  2. \(\ln{x}+\ln{0.2}=\ln{e}\)

  3. \(\ln{4x^2}-\ln{16}=\ln{4}-\ln{2x}\)

Revision of Differentiation and Integration Techniques

Differentiate the following functions:

  1. \(x(7x+8)^{2}\)

  2. \(\dfrac{5x^{2}}{(2x^{3}+4)^{4}}\)

Integrate the following functions:

  1. \((2x+1)\ln(x+1)\)

  2. \(x e^{x^{2}}\)

Multiple Choice Questions

  1. If \(f(x)=(x^2-3x)^6(4-x)^5\), then \(f'(x)\) is equal to:

\(\text{(A) } (x^2-3x)^5(4-x)^4(5x^2-21x+24)\)

\(\text{(B) } -(x^2-3x)^5(4-x)^4(7x^2-51x+72)\)

\(\text{(C) } -(x^2-3x)^5(4-x)^4(17x^2-81x+72)\)

\(\text{(D) } 6(x^2-3x)^5(2x-3)(-5)(4-x)^4\)

\(\text{(E) none of these}\)

2) If \(y=e^x \ln{x}\), then \(\dfrac{dy}{dx}\) is equal to:

\(\text{(A) } \dfrac{e^x(1+x\ln{x})}{x}\)

\(\text{(B) } e^x(x+\ln{x})\)

\(\text{(C) } xe^x\)

\(\text{(D) } \dfrac{x}{e^x}\)

\(\text{(E) none of these}\)

3) If \(\dfrac{500}{12+5e^{-0.5x}}\), then \(\dfrac{dy}{dx}\) is equal to:

\(\text{(A) } 500(-1)(12+5e^{-0.5x})^{-2}\)

\(\text{(B) } 500(-1)(12+5e^{-0.5x})^{-2}(12+5e^{-0.5x})\)

\(\text{(C) } 500(-1)(12+5e^{-0.5x})^{-2}(-2.5e^{-0.5x})\)

\(\text{(D) } 500(-1)(5(0.5)e^{-0.5x})^{-2}\)

\(\text{(E) none of these}\)

4) \({\displaystyle \int \dfrac{x^2+4x-\sqrt{x}}{x^2} dx}\), expressed in terms of an arbitrary constant \(c\) is equal to:

\(\text{(A) } x+\ln{4x}-\frac{2}{3}x^{\frac{3}{2}}+c\)

\(\text{(B) } x+\ln{4x}+\dfrac{2}{\sqrt{x}}+c\)

\(\text{(C) } x+4\ln{x}+\dfrac{1}{2\sqrt{x}}+c\)

\(\text{(D) } x+4\ln{x}+\dfrac{2}{\sqrt{x}}+c\)

\(\text{(E) none of these}\)

5) \({\displaystyle \int \left(\dfrac{3x}{2}-\dfrac{9}{4}\right) e^{x(x-3)} dx}\), expressed in terms of an arbitrary constant \(c\) is equal to:

\(\text{(A) } \dfrac{3}{4}e^{x^2-3x}+c\)

\(\text{(B) } e^{x^2-3x}+c\)

\(\text{(C) } \dfrac{9}{4}e^{x^2-3x}+c\)

\(\text{(D) } \dfrac{3}{2}e^{x^2-3x}+c\)

\(\text{(E) } \dfrac{4}{3}e^{x^2-3x}+c\)

6) \({\displaystyle \int (3x+1)^{2} e^{-2x} dx}\), expressed in terms of an arbitrary constant \(c\) is equal to:

\(\text{(A) } -0.5(3x+1) e^{-2x}+1.5(3x+1)^{2} e^{-2x}-2.25 e^{-2x} + c\)

\(\text{(B) } -0.5(3x+1)^{2} e^{-2x}+1.5(3x+1) e^{-2x}+2.25 e^{-2x} + c\)

\(\text{(C) } -0.5(3x+1)^{2} e^{-2x}-1.5(3x+1) e^{-2x}-2.25 e^{-2x} + c\)

\(\text{(D) }-0.5(3x+1)^{2} e^{-2x}+1.5(3x+1) e^{-2x}-2.25 e^{-2x} + c\)

\(\text{(E) none of these}\)

Tutorial 2: ODE’s and Classifcation

Outcomes: At the end of this tutorial you should be able to

- Understand the meaning of an ordinary differential equation.
- Know why classification of an ODE is useful.
- Understand the ways that an ODE can be classified.
- Classify a given differential equation.

Conceceputal Questions

  1. Explain the ways in which one can classify a differential equation.

  2. What does the degree of a differential equation inform us about its linearity?

Classifcation of ODE’s

Classify the following differential equations by checking first if they are ODE’s, and then in terms of their order, linearity, homogeneity, and coefficients. Give reasons for your answers

  1. \(t \dfrac{d^3 x}{dt^3}-2\left(\dfrac{d x}{dt}\right) ^4+x=0\)

  2. \(y y'+2y=1+x^2\), where \(y=y(x)\)

  3. \(y''+9y=\sin y\), where \(y=y(x)\)

  4. \(\dfrac{d^2 R}{dt^2}=\dfrac{\kappa}{R^2}\), where \(\kappa\) (read: kappa) is a constant

  5. \(x^5\dfrac{d^4 y}{dx^4}-x^3 \dfrac{d^3 y}{dx^3}+6y=0\)

  6. \(\dfrac{d^2 x}{dt^2}=\sqrt{1+\left( \dfrac{d x}{dt}\right) ^2}\)