Calculus: Semester 1 Exam Review

  • Analysis of Functions - be able to use your graphing calculator to graph any function and answer questions about the graph/function. Describe domain and range using interval notation.
  • Limits (1 and 2 sided)
  • calculate limits using algebra
  • substitution
  • factor, cancel and then substitute
  • for limits ,know and look at degrees of numerator/denominator
  • (and variations of them)
  • estimate limits from graphs (be careful of hidden behavior) - be especially concerned with asymptotes
  • Continuity
  • Defn of continuity: f(x) is continuous iff for all values of x in the domain (the limit from the left and right must be same AND the function must = that same value)
  • The Extreme Value Thm: if f(x) is continuous at every pt in the closed interval [a,b] then f(x) has both a max and a min value somewhere in [a,b]
  • Discontinuities: jump, hole, asymptote, oscillating (be able to answer questions about removable discontinuities-holes)
  • Derivatives
  • Average Rate of Change = , vs. Instantaneous Rate of Change (slope of tangent to curve—1st derivative)
  • Defn. of derivative:
  • differentiability vs.continuity (a function will be differentiable if the function is continuous AND there are no abrupt turns (corners, cusps, Vert. slope at a pt, etc) within the interval ---- a function can have a one-sided derivative
  • Mean Value Thm - If f(x) is a continuous function on [a,b] then there is at least one pt,C, where (wherethe instantaneous slope of the curve = the average slope of the curve) --- ex, if avg speed between pt a and b is 75 mph, then at least once between a and b you were travelling 75 mph.
  • derivative rules (see attached)
  • powers of x, trig functions, exponential and log functions (base e and other), inverse trig functions
  • derivatives of sums,products and quotients, and chain rule
  • implicit diff. - differentiating functions with x and y terms mixed together
  • Application of Derivatives
  • graphical relationships between f, f’ (incr./decr. or critical pts.), f’’ (concavity & pts.of inflection)
  • optimization
  • related rates problems

Derivative Rules:

  • powers of x:
  • trig functions:
  • exponential functions: , where u is a function of x

, where u is a function of x

  • log functions: , where u is a function of x

, where u is a function of x

  • derivatives of sums: differentiate each term and add
  • derivatives of products:
  • derivatives of quotients:
  • chain rule: If y is a function of u and u is a function of x,
    then
  • implicit differentiation – finding dy/dx in an eqn. with x and y terms mixed together:
  • differentiate both sides with respect to x (use chain & product rules as needed),
    If differentiating an x term, just find the derivative, but if differentiating a y term, find the derivative with respect to y and multiply by dy/dx
  • collect terms with dy/dx on one side
  • factor out dy/dx
  • solve for dy/dx by dividing