Calculus 1 Review

• LIMITS- direct substitution


  Directly plugging in the value "x" is approaching.
• Perfect squares


  [(x^2)-9]=[(x+3)(x-3)]~essentially taking the square of both parts
• Factoring trinomials

  [(x^2)+2x-15](x^2)=a(2x)=b(-15)=c

  ~find two values that add to "B" and multiply to "C"

  ~Then plug in the new values for b (ensure to add the variable)[(x^2)+(5x)-(3x)-(15)]

  
  

  ~separate the values (think that's too many dang numbers chop in half)

  [(x^2)+(5x)][(-3x)-(15)]

  
  

  ~Factor each half

  {x} [x+5] {-3}[x+5]

  
  

  ~Now put the factor variables together and use one of the like factor (or equations or whatever)

  
  

  [x+5][x-3]

  
  

  ~That's the answer goofball
• Derivative basics (how to read)

  when a function starts with d/dx that means the derivative of said function with respect to "x"
• Derivative: Power Rule

  d/dx[x^n]= [(n*x)]^[(n-1)]
• Derivative rewrite for (3/x)


  (3)x^-1
• Derivative rewrite for (sqrt(x))


  x^(1/2)
• How to solve derivatives

  1) rewrite the expression

  2) use your rules to take the derivative

  3) simplify and remove any negative exponents
• Continuity of piecewise function: solving for c


  1) Set the pieces equal to each other

  2) plugin for the known variable

  3) solve for the unknown variable
• Derivative rules: Exponential functions: {d/dx}[(e)]^[(u)]=


  =[(e)^u] * [U']
• Derivative rules: logarithmic natural log functions: {d/dx} Ln(u)


  =[(u')/(u)]
• Derivative rules: power rule: {d/dx}(f * g)


  =(f' g) + (f g')
• Derivative rules: constants:{d/dx}(c)


  =0

  
  

  The slope is equal to zero (straight line)
• Derivative rules: {d/dx}(x)


  =1x^1-1=x^0=1
• Derivative rules: {d/dx} (ax)

  =ax^1-> 1a= a
• Derivative rules: Exponential: {d/dx} [a^x]


  =[ ln(a)]*[(a)^x]
• Derivative rules: power rule: {d/dx} log (x) a


  =[1]/[x (ln(a))]
• Derivative rules: Trig: {d/dx} [Sin(x)]

  =cos(x)
• Derivative rules: Trig: {d/dx} [Cos(x)]

  -sin(x)
• Derivative rules: Trig: {d/dx} [Tan(x)]


  [sec^2(x)]
• Derivative rules: Inverse Trig: {d/dx} [Sin^-1(x)]

  [1]/[√(1−x^2)]
• Derivative rules: Inverse Trig: {d/dx} [Cos^-1(x)]

  [−1]/[√(1−x^2)]
• Derivative rules: Inverse Trig: {d/dx} [Tan^-1(x)]


  [1]/[(1+x^2)]
• Derivative rules: Multiplication by constant: {d/dx} [c*f]


  =(c)*(f')
• Derivative rules: Sum Rule: {d/dx} [f + g]


  [f' + g' ]
• Derivative rules: Difference : {d/dx} [f-g]


  [f' - g']
• Derivative rules: Quotient Rule : {d/dx} [(f)/(g)]


  [(f' g)- (g' f)]/[(g)^(2)]Note: Low D High - High D Low/ Square the bottom and away we go!!!
• Derivative rules: Reciprocal rule : {d/dx} [(1)/(f)]


  [ -f' ]/ [(f)^(2)]
• Derivative rules: Chain Rule : {d/dx} [f º g]Also written as [f(g(x))]


  (f' º g) × g' Also written as{d/dx} [f(g(x))]= [F'(g)] * [g']
• So, you're given:

  lim x->3

  [(x^2)+(2x)-(15)] / [(x^2)-(9)]

  ~Factor: do this so that you can cancel out the problem.

  Iim x->3

  [(x+5)(x-3)]/[(x+3)(x-3)]

  Notice: you can cancel (x-3)

  Now you have

  lim x->3 [(x+5)]/[(x+3)]

  ~plug and chug

  1)

  lim x->3

  [(x+5)]/[(x+3)]

  2)

  [({3}+5)]/[({3}+3)]

  3)

  =[(8)]/[(6)]

  4) simplify as needed:

  =[(4)]/[(3)]