Calculus Final Review Test 1 (Unit 1-3)

• Secant Line

  A line that passes through two points on a point, same thing as the average rate of change
• Slope of Tangent Line

  Derivative at given value
• Secant line is equal to

  the average rate of change
• Slope of Secant Line

  f(b) - f(a) / b-a
• Tangent Line

  A line that passes through one point on a curve, same thing as the instantaneous rate of change
• Equation of a tangent line

  y-y1 = m(x-x1)
• Use the slope of secant lines to

  find the slope of a tangent line
• Tangent line is equal to

  The instantaneous rate of change
• When finding the instantaneous rate of change

  You can choose your own intervals to find the average rate of change
• The Limit of a function

  When the y-value on both sides of a function is approaching a value at a given x-value
• The Left-Hand Limit

  "limit from below" the limit as x goes to "a" from the left ( or below), x < a
• The right-hand limit

  "limit from above" the limit as x goes to "a" from the right ( or above), x>a
• The full limit

  If and only if both the left and right hand limits meet at the same point
• If both sides of the limit do not equal to one another, it means the limit

  DNE
• Limit of a Function (informal definition)

  We can get as close to the values of "L" as we want by taking x-values closer and closer ( but NOT equal to a)
• To find the limit of a function, you can...

  make a table and see the trend of what the values of y are approaching to
• Lim as x approaches positive/negative infinity

  It means that we get very large negatively or positive as the output
• Limit as x approaches positive infinity

  If and only if both the left and right limits are BOTH positive infinity
• Infinity is

  NOT a number
• When we have a limit that goes to plus or minus infinity,

  there is a vertical asymptote
• The Squeeze Theorem

  If f(x) ≤ g(x) ≤ h(x) when x is near "a" and limx→a f(x) = limx→a h(x) = L, then lim x→a g(x) = L

  
• Limit Laws

  lim (f±g) = lim f ± lim g

  lim (c ⋅ f) = c ⋅ lim f

  lim (fg) = lim f ⋅ lim g

  lim (f/g) = lim f / lim g for lim g ≠ 0

  lim √f(x) = √lim f(x)

  lim c = c

  lim xⁿ = aⁿ

  lim x = a