Topics for review

Review 1

Calculus I

• Limits
• Derivative rules: power, product, quotient, chain, \(\ln(x)\), \(e^x\)
• Fundamental theorem of calculus
• Integration with substitution

Applications

• Area between two curves
• Volume using cross-sections
• Linear density and mass
• Average value of function
• Mean value theorem for integration
• Volumes of revolution (x-axis, y-axis, other)
  • Disk method
  • Washer method (or disk subtraction)
  • Cylindrical shell method
• Work = Force * Distance
  • \(\text{Work} = \displaystyle\int_{a}^{b} \text{Force}(x) \, dx \)
  • \(L(y) = (\text{density} \cdot \text{area of layer}) \cdot (\text{acceleration due to gravity}) \cdot (\text{vertical distance lifted}) \)
  • Work against gravity \( = \displaystyle\int_{a}^{b} L(y) \,dy \)

Methods

• Integration by parts
• Trigonometric integrals
  • \(\sin^n(x)\) where \(n\) is odd
  • \(\cos^m(x)\) where \(m\) is odd
  • \(n\) or \(m\) even
  • \(\sin(Ax) \cdot \cos(Bx)\)

Samples

• Work: Calculate the work against gravity (9.8 m/s/s) required to build a right circular cone of height 4m and base radius 1.2 m with density 600 kg/m^3.
  
• Area: \( y=\dfrac{x}{5}, y=5x, y=(x-5)^2 + 1 \) for x in [0, 5]
  
• Volume: Region between \( y=x^4 \) and \( y=8x \) revolved around \( x = 2 \).
  
  

Review 2

Previous material and …

Integration methods

• Trigonometric substitutions for \( \sqrt{a^2 - x^2}, \sqrt{x^2 + a^2}, \sqrt{x^2 - a^2} \)
• Partial fractions (with linear factors)
• Long division for improper rational functions
• Improper integrals
• Numerical integration
  • midpoints
  • trapezoids
  • Simpson's rule

More applications

• Probability density functions
• Arc length
• Surface area
• Centroids

Samples

• Centroid: Region defined by \(y = 2-x^2, y=x^3, x=0 \), which has intersection at \( (1,1) \).
  
• Partial fractions: \( \displaystyle\int \frac{x^2 + x + 1}{x^3 - 4x} \,dx \)
  
• Use Simpson's rule (n=6) to approximate the surface area of \( f(x) = x \cdot \ln(x) \) on the interval \( [1, 3] \) rotated around the x-axis.
  

Review 3

Previous material and …

Differential equations

• Order: first, second, etc.
• Separable
• Initial conditions
• y' = k(y - b) form
• Newton's law of cooling
• Slope fields
• Euler's method
• First-order linear

Sequences and series

• Sequences
• Partial sums
• Telescoping series
• Geometric series
• nth term divergence test
• Integral test
• p-series
• Harmonic series
• Direct comparison test
• Limit comparison test
• Alternating series test
• absolute value of series converges implies series converges
• Converges absolutely vs converges conditionally
• Root test
• Ratio test
• Power series
• radius of convergence
• interval of convergence
• Taylor polynomials
• Taylor series
• Maclaurin series for \( e^x = \displaystyle\sum_{n = 0}^{\infty} \dfrac{x^n}{n!} \)

Samples

See series practice: 1, 2

Review for final exam

Previous material and …

Sequences and series

• Maclaurin series for \( \sin(x) = \displaystyle\sum_{n = 0}^{\infty} (-1)^n \dfrac{x^{2n+1}}{(2n+1)!} \)
• Maclaurin series for \( \cos(x) = \displaystyle\sum_{n = 0}^{\infty} (-1)^n \dfrac{x^{2n}}{(2n)!} \)