![]() Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration Pre-Calculus Review.Use of implicit differentiation to find the derivative of an inverse function Pre-Calculus Review.Modeling rates of change, including related rates problems Pre-Calculus Review (Special Review of this Topic).Optimization, both absolute (global) and relative (local) extrema Pre-Calculus Review (Optimization is NEW).Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration Chapter 10/11.Analysis of curves, including the notions of monotonicity and concavity Pre-Calculus Review.Points of inflection as places where concavity changes Pre-Calculus Review.Relationship between the concavity of ƒ and the sign of ƒ' Pre-Calculus Review.Corresponding characteristics of the graphs of ƒ, ƒ', and ƒ'' Pre-Calculus Review.Verbal descriptions are translated into equations involving derivatives and vice versa. The Mean Value Theorem and its geometric interpretation Pre-Calculus Review (MVT is NEW).Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ' Pre-Calculus Review.Corresponding characteristics of graphs of ƒ and ƒ' Pre-Calculus Review.Approximate rate of change from graphs and tables of values Pre-Calculus Review.Instantaneous rate of change as the limit of average rate of change Pre-Calculus Review.Tangent line to a curve at a point and local linear approximation Pre-Calculus Review.Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. Relationship between differentiability and continuity Pre-Calculus Review.Derivative defined as the limit of the difference quotient Pre-Calculus Review.Derivative interpreted as an instantaneous rate of change Pre-Calculus Review.Derivative presented graphically, numerically, and analytically Pre-Calculus Review.The analysis of planar curves includes those given in parametric form, polar form, and vector form. Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem) Pre-Calculus Review (IVT is NEW).Understanding continuity in terms of limits Pre-Calculus Review.(The function values can be made as close as desired by taking sufficiently close values of the domain.) Pre-Calculus Review An intuitive understanding of continuity.Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth) Pre-Calculus Review.Describing asymptotic behavior in terms of limits involving infinity Pre-Calculus Review.Understanding asymptotes in terms of graphical behavior Pre-Calculus Review.Estimating limits from graphs or tables of data Pre-Calculus Review. ![]() Calculating limits using algebra Pre-Calculus Review.An intuitive understanding of the limiting process Pre-Calculus Review.Limits of functions (including one-sided limits) The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function. Analysis of graphs With the aid of technology, graphs of functions are often easy to produce.
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