Sketching_graphs_using_calculus
Combine the tools provided by differential calculus with other algebraic tools in order to obtain detailed sketches of various functions.
Review_Increasing_decreasing_intervals_extrema
Review your understanding of increasing and decreasing intervals, and extremum points, with some challenge problems.
Solve geometrical and real-world problems that involve finding the maximum (or minimum) possible value of a quantity. For example, find the maximum area of a rectangle whose perimeter is given.
Concavity describes the shape of a graph as it increases or decreases: a graph that's concave up is shaped like a cup, U, and a graph that's concave down is shaped like a cap, ∩. Learn more about concavity and how it relates to a function's second derivative.
Analyze functions to find their relative extrema (i.e. their relative minimum and maximum points).
Solve problems about motion along a line using the power of differential calculus. For example, given the position of a particle as a function of time s(t), find the particle's maximum velocity.
The method of linear approximation (also called local linearization) allows us to approximate a function at hairy x-values using the line tangent to the function's graph at strategic points.
The mean value theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in (a,b) such that f'(c) is equal to the function's average rate of change over [a,b].
Increasing_decreasing_intervals
Analyze functions to find the intervals in which they are increasing or decreasing.
Solve various rate of change problems where a real-world situation is modeled by an algebraic function.
Solve geometrical and real-world problems that concern multiple quantities that change at different, but related, rates. For example, given the rate of change of a circle's radius, find the rate of change of the circle's area.
L'Hôpital's rule provides us with an easy, almost magical way of finding indeterminate limits of quotients of functions using the functions' derivatives. In short, the rule says that if the limits of functions f and g at x=a are 0 (or ထ) and the limit of f'(x)/g'(x) at x=a is equal to L, then the limit of f(x)/g(x) at x=a is also equal to L.
Critical points are points where a function may obtain their minimum or maximum value. They play a critical role (pun intended) in analyzing the increasing and decreasing intervals of functions, and in finding their minimum and maximum points.
Points of inflection are points where the function changes concavity. Learn how to find and analyze them.
Review_Concavity_points_of_inflection
Review your understanding of concavity and points of inflections with some challenge problems.
Solve problems about motion on a 2-dimensional plane using the power of differential calculus. For example, given the (x,y) position of a particle as a function of time (x(t),y(t)), find the particle's position when its acceleration is 0.
Review_Derivative_applications
Review your understanding of the various applications of differential calculus with some challenge problems.
Analyze functions to find their absolute extrema (i.e. their absolute minimum and maximum points).
Derivative_applications_skill_check
Test your understanding of some of the most useful applications of the derivative.
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