Unit 5: Derivatives 4.1 The Derivative Function State what the derivative represents (3 interpretations) determine expression for from first principles evaluate at x = a Differentiability sketch the graph of the derivative, using the slopes of the tangent lines at various places on the curve of f. Distinguish between continuity and differentiability. Determine whether a function is differentiable at x = a: algebraically or graphically. 4.2 Derivatives of Polynomial Functions Derivative Rules: Constant, Constant Multiple, Power, Sum and Difference Be able to state and apply the rules 4.3 The Product Rule state and prove the product rule. Apply the product rule the power of a function rule Equations of Tangents and Normals Find the equation of the tangent: To a curve at a given point. Parallel to a given line Find the equation of a normal to a curve Find points on a curve where the tangent has a given slope. 4.4 The Quotient Rule the quotient rule. Apply the quotient rule. Find the equation of a given tangent Find points on the graph where the tangent has a given slope. 4.6 The Derivative of a composite Function differentiate composite functions the Chain Rule and power-product-quotient-chain rules combined Newton’s Method: differentiate composite functions using the Chain Rule Leibniz’s Method: Unit6: Applications of Derivatives 1. 5.2 Higher-Order Derivatives, Velocity, and Acceleration Find higher order derivatives use position function s(t), to problems involving velocity and acceleration using the fact that: 2. 5.4 Maximum and Minimum on an Interval Extreme values on a closed interval using conditions set to find critical numbers and function values at the endpoints of an interval to determine the extrema (max./min. values) on that interval apply to graphs and problem solving 3. 5.5 - 5.6 Optimization Problems apply problem solving steps to maximizing and minimizing problems involving: area, volume, geometry, economics, social sciences and science 4. The Derivatives of the Sine and Cosine Functions Know the derivatives of the sine and cosine function. apply the derivatives of sine/cosine functions to differentiate other related functions. 5. Applications of Trigonometric Function solve rate of change problems and applications involving trigonometric functions.
Unit 7: Derivatives of Exponential and Logarithmic Functions 1 .8.1 Derivatives of Exponential Functions - define e - Be able to apply the derivative of 2. 8.2 The Derivative of the Natural Logarithmic Function - know the laws or logarithms and be able to simplify functions using the laws. - define the natural logarithm - Be able to prove and apply the derivative of y = ln x 3. 8.3 Derivatives of General Exponential and Logarithmic Functions - Be able to apply the derivatives of 4. 8.4 Optimization Problems - the process of optimization with exponential and logarithmic functions. Unit 8: Curve Sketching Review Prerequisite Skills: - solving inequalities - division of polynomials Domain Determine the domains for of functions that will be analysed in this unit. Including rational and radical expressions. 9.1 Increasing and Decreasing Functions - define when functions are increasing or decreasing on an interval. - state conditions on the first derivative for functions to be increasing or decreasing - use conditions to determine where a function is increasing or decreasing. 9.2 Critical Points, Relative Maxima, and Relative Minima - state conditions for: -absolute maximum or minimum values - relative maximum or minimum values - critical numbers - important numbers - use the conditions to determine relative maxima or minima, critical and important numbers for various functions. 9.3 Vertical, Horizontal - define: infinite limits, one-sided infinite limits, finite limits at infinity - state conditions for rational functions to have vertical, horizontal and linear oblique asymptotes - determine the vertical, horizontal, or oblique asymptote(s) of a rational function and the behaviour of the function close to the asymptote. 9.3 Oblique Asymptotes - outline a process to find oblique asymptotes and apply 9.4 Concavity and Points of Inflection - work through investigations on p. 363 - summarize conditions for concavity, the second derivative test and points of inflection 9.5 An Algorithm for Graph Sketching - outline steps for analysis and curve sketching - examples of polynomial, rational, exponential and logarithmic functions.