Calculus: Semester 1 Exam Review
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