Exam 2 Review Topics

 

(Note: LíHopital Rule will not be covered by Exam 2)

 

Topics: Derivatives, chain rule, implicit differentiation, related rate, equation of tangent line, tangent line approximation, linear approximation, critical points (2 types), inflection points, local extrema, absolute extrema in closed intervals, absolute extrema in unbounded intervals, intervals of increasing and decreasing, intervals of concave down and concave up, optimization, story problem of optimization, mean value theorem, families of functions.

 

Techniques: All differentiation rules: summation rule, scalar product rule, product rule, quotient rule, chain rule, implicit differentiation, related rate problem. Derivatives of inverse functions, derivatives of inverse trigonometric functions. Linear approximation, equation of tangent line. Finding critical points, finding absolute extrema on intervals, 1st derivative test for local extrema, 2nd derivative test for local extrema. Determining the signs of derivative functions over intervals segmented by critical points, determining increasing and decreasing intervals of functions, technique of one-point testing. Determining the signs of the second derivatives of functions, determining the interval of concave down and concave up and points of inflection. Sketch graphs of inverse functions. Sketch graphs of functions featuring x- and y-intercepts, vertical and horizontal asymptotes, local extrema, intervals of increasing and decreasing, intervals of concave down and concave up, and points of inflection. Mean Value Function Theorem. LíHopitalí rule for 0/0 type, ∞/∞ type, 0*∞ type, 1^∞ type. Story problems of optimization. Determining the signs of the second derivatives of functions using one-point testing method, determining the interval of concave down and concave up and points of inflection, 2nd derivative test for local extrema. Curve sketching: horizontal asymptotes, vertical asymptotes, intervals of increasing and decreasing, intervals of concave up and concave down, local maximum and local minimum points, inflection points. Story problems in optimization: decision variable, objective function, and constraint. Formulate optimization problems and solve them.

 

More Techniques: Quadratic formula for roots, factorization of a^2-b^2, a^3-b^3, multiply and divide to maintain and transform quantities. Basic forms and shapes of trig and inverse trig functions, exponential functions, logarithmic functions, and polynomials: linear function, parabola, cubic polynomials. Definitions of trigonometric functions. Definitions of inverse trigonometric functions and exponential functions (logarithmic functions). Basic rules, identities, and limiting properties as x → ∞ for exponential and logarithmic functions. Special values of trigonometric functions and exponential and logarithmic functions. Derivative of power functions, exponential functions, logarithmic functions, trigonometric functions and their inverses. Antiderivatives of power functions, exponential functions, trigonometric functions and their inverses.

 

Calculator Skills: Sketch graphs, tracing intersection points and roots, finding numerical limits

 

Partial Review Problems: All homework, quiz, lecture example problems, and sample exams. Repeat the problems until you can do them without any help.