Calculus Final Review Test 1 (Unit 6-8)

• If r > 0 is a national number (integer/integer) then the limit

  Lim as a goes to infinity 1/x^r =0
• If r > 0 is a rational number such that x^r is defined for all x then the limit

  Lim as x goes to negative infinity 1/x^r =0
• Equation of a tangent line

  y - y0 = m(x- x0)
• Derivative of a function

  f’(x) = lim h —> 0 ( f(x+h) - f(x) / (h))

  
• Infinity is

  NOT a number, you may not do arithmetic with it
• Lim as a approaches infinity f(x) = x^3

  Infinity (it increases quickly)
• Lim as x approaches negative infinity f(x) = 1/x

  Zero ( it gets smaller and smaller)
• Lim as x approaches infinity f(x) = cos(x)

  DNE (oscillates too much)
• How to evaluate limits at infinity

  If p(x) and q(x) are polynomials, then to evaluate Lim as a goes to positive/negative infinity p(x)/q(x), we divide the numerator and denominator by the highest power of x in the denominator
• Horizontal Asymptote

  Line y=L of y=f(x) is a horizontal asymptote if either Lim as x approaches infinity = L or Lim as x approaches negative infinity = L
• Average rate of change

  Slope of secant line through the points (a, f(a)) and (b,f(b))
• Instantaneous rate of change

  Slope of tangent line through one point (a,f(a))
• Derivative of a function at x=a

  f'(a)= lim as h approaches 0 ((f(a+h) - f(a)) / h)
• Derivative at a point x=a

  Instantaneous rate of change, slope of the tangent line at x=a
• The derivative is the slope of a tangent line at a point
• f(x) horizontal slope = 0

  f'(x) intersects at 0

  
• f(x) positive to negative

  f’(x) minimum point
• f(x) negative to positive

  f’(x) maximum point
• f(x) negative/decreasing slope

  f’(x) line is below the x-axis (negative)
• f(x) positive/increasing slope

  f’(x) is above the x-axis (positive)
• When the function f(x) is increasing

  the derivative f’(x) must be positive
• When the function f(x) is decreasing

  The derivative f’(x) must be negative
• The function f(x) is NOT differentiable when there are

  holes (DNE)

  corners ( not approaching the same value)

  Cusps ( going to infinity)

  Vertical lines

  Jump discontinuities
• If a function is differentiable at x=a

  it is continuous
• If a function is continuous at x=a

  it is NOT necessarily differentiable
• First Derivative

  f’(x) = y’ = dy/dx = d/dx(f(x))
• Second Derivative

  f’’(x) = y’’ = d^2y/dx^2 = d^2/dx^2 (f(x))
• f(x), f’(x), f’’(x)

  Position, Velocity, Acceleration
• If r > 0 is a national number (integer/integer) then the limit

  Lim as a goes to infinity 1/x^r =0
• If r > 0 is a rational number such that x^r is defined for all x then the limit

  Lim as x goes to negative infinity 1/x^r =0
• Equation of a Tangent line

  y - y0= m(x- x0)