Calculus 2 Final Review ( Terms and formulas)

• Find the area of the region bounded by the given curves - steps

  1. Find the intersection points ( equal the equations to one another )
  2. Figure out which curve is above/below, above curve will go first in the equation
  3. Find the area: int from a to b ( F(x) - G(x)) dx, simplify the equation and solve
• Find the Area of the region between two curves example

  y = x^2 - 6x, y = 5x + 12

  1. Equal the two equations, x^2 - 6x = 5x + 12, two intersection points are x = -1 and x =12
  2. Which curve is above? y = 5x + 12
  3. Set up the equation: int from -1 to 12 ( 5x + 12 ) - ( x^2 - 6x) dx
  4. Simplify and solve, final answer is 2197/6
• Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line

  y = ln(x), y = 1, y = 5, x = 0, about the y-axis

  1. Draw a picture of the graph and lines
  2. Need to covert y = ln(x) to an x equals, so it would be x = e^y
  3. If it is rotating around the y-axis, it will be in terms of y. So the equation would be, V = pi int from 1 to 5 (e^y)^2 dy
  4. Use a u-sub to solve the equation: e^2y so u = 2y
  5. Integrate and solve, pi ( 1/2e^2y) from 1 to 5
• Equation to find the Volume of a solid rotated around the x-axis

  V = pi * int from a to b (f(x))^2 dx
• Equation to find the Volume of a solid rotated around the y-axis

  V = pi * int from a to b (g(y))^2 dy
• Logarithmic Differentiation: How to Solve

  1. Take the natural log of both sides of the equation ( ln(x) )
  2. If a side of your function is raised to a power, make sure you move the power down to the left side.
  3. Differentiate implicitly, using dy/dx
  4. Goal is to solve for dy/dx and then make sure to substitute your y value back into the final answer.
• Logarithmic Differentiation Example: y = x^8x

  1. Take the natural log both sides: ln(y) = ln(x)^8x
  2. Move the 8x down to the left side: ln(y) = 8x ln (x)
  3. Differentiate both sides: d/dx(ln(y)) = d/dx ( 8x ln(x))
  4. Simplify: (1/y) * dy/dx = 8 ln(x) + 8 ( product rule )
  5. Multiply both sides by y: dy/dx = y( 8 ln(x) + 8)
  6. Fill in y for final answer: dy/dx = x^8x ( 8 ln(x) + 8)
• Trig Integrals to Remember!

  What is the derivative of sin^-1(x)?


  1/sqrt ( 1 - x^2 )
• Trig Integrals to Remember!

  What is the derivative of tan^-1(x)?

  1/(x^2+1)
• Derivative of Inverse Trig - Sine, y = 4 sin^ -1 (x^2)

  1. Recall that sin^-1(x) has the derivative of 1/sqrt(1 - x^2). Use the information to fill in for the function above.
  2. So, y' =( 4/ sqrt( 1 - (x^2)^2) ) * 2x , make sure to get the derivative of whatever is inside of the inverse function. So, derivative of x^2 is 2x.
  3. Final answer: y' = 8x/ sqrt( 1 - x^4)
• Product Rule

  f'(x)g(x) + g'(x)f(x)
• Quotient Rule

  ( f'(x)g(x) - g'(x)f(x) )/ g(x)^2
• Chain Rule

  d/dx ( f(g(x)) = f'( g(x) ) * g'(x)
• Chain Rule Example: f(t) = cosh^2(5t^2+2)

  1. First, move the squared around the whole function: (cosh(5t^2+2))^2
  2. Now, differentiate as usual: 2(cosh(5t^2+2)) * sinh(5t^2+2 )
  3. Take the derivative of the inside of the sinh function: sinh(5t^2+2) * 10t
  4. Combine all elements: 20t cosh(5t^2 + 2) sinh( 5t^2 + 2)
• L'Hopital's Rule

  When the lim as x approaches a (f(x)/g(x)) = 0 or the lim as x approaches a (f(x)/g(x)) = infinity, we have lim as x approaches a (f(x)/g(x)) = lim as x approaches a (f'(x)/g'(x))
• L'Hopital's Rule Example: lim as x approaches infinity (ln(x))^4/6x^3

  1. Differentiate the top function as f'(x) and differentiate the bottom function as g'(x)
  2. f'(x) = 4 (ln(x))^3/x and g'(x) = 18x^2
  3. Now, fill it back into limit form: 2/9 lim as x approaches infinity of (ln(x)/x)^3. The function (ln(x)/x) converges to 0, so 2/9 * 0 is 0
  4. Final answer: the lim as x approaches infinity (ln(x))^4/ 6x^3 converges to 0.
• Integration by Parts Formula

  int ( udv ) = uv - int ( vdu )

  
• $\int_{ }^{ }e^{-x}dx\ =$$\ -e^{-x}\ +$$\ C$
• int ( sin(x) ) = - cos(x) + C
• int ( cos(x) ) = sin(x) + C
• Improper Integrals

  Type 1: Interval is infinite

  Type 2: F(x) is NOT bounded ( discontinuous at a point)

  • Convergent if you get a number, divergent if you get infinity
• Geometric Series

  If |r| < 1, the series converges

  If |r| > or = to 1, the series diverges
• P-Series

  If p > 1, the series converges

  If p < or equal to 1, the series diverges
• Ratio Test

  Use when there is factorials or there are variables raised to the n-power
• Root Test

  Use when an is all raised to the n-power

  ex: (5x/3n+1) ^ n

  
  

  If the answer is < 1, then it converges

  If the answer is > 1, then it diverges
• If there is lim as n approaches infinity (1/n+1), the series will converge to 0.

  If it is equal to 0, the Radius of Convergence is infinity and the Interval of Convergence is (-infinity, infinity ).
• () means Diverge

  [] means Converge
• When finding the interval of convergence, don't forget that the x equation will always be between -1 < |x| < 1
• lim as n approaches infinity of (1 + 1/n)^n is equal to e
• Ratio Test

  If the answer is < 1, the series converges

  If the answer is > 1, the series diverges