3.5 Derivatives of Trigonometric Functions
MTH 251 - CALCULUS
FINAL EXAM REVIEW
3.5 Derivatives of Trigonometric Functions
• Be able to determine the derivatives of all 6 of the trigonometric functions.
• Be able to find the derivative of a function involving trigonometric, polynomial and exponential functions using the product and quotient rules.
• Be able to find the equation of the tangent line to a function at a point.
• Be able to evaluate limits involving trigonometric functions.
3.6 Derivatives as Rates of Change
• Be able to solve applied problems involving velocity and acceleration.
• Be able to solve applied problems involving other rates of change.
3.7 The Chain Rule
• Be able to separate a composite function into its inner and outer functions.
• Be able to evaluate a derivative using either version of the chain rule.
• Be able to calculate derivatives using the chain rule and the derivative rules learned in the previous sections.
• Be able to find the equation of the tangent line to a function at a point.
• Be able to calculate the derivative of a composite function given information about the inner and outer functions and their derivatives.
3.8 Implicit Differentiation
• Be able to find the derivative of a function both explicitly and implicitly and show that they are equal.
• Be able to find the equation of the tangent line to a function at a point using implicit differentiation.
• Be able to calculate the second derivative of a function using implicit differentiation.
3.9 Derivatives of Logarithmic and Exponential Functions
• Be able to calculate the derivative of functions involving logarithms and trigonometric, polynomial and exponential functions.
• Be able to find the equation of the tangent line to a function involving logarithms, at a point.
• Be able to use logarithmic differentiation to find the derivative of a function.
3.10 Derivatives of Inverse Trigonometric Functions
• Be able to calculate derivatives involving inverse trigonometric functions.
• Be able to find the equation of the tangent line to a function involving inverse trigonometric functions at a point.
3.11 Related Rates
• Be able to find the derivative of a function taken with respect to time.
• Be able to solve applied related rates problems. Remember: Do not plug in any non-constant values until after taking the derivative with respect to time.
4.1 Maximum and Minimum Values
• Be able to determine local and absolute maximums and minimums for a function given a graph of the function.
• Be able to sketch the graph of a function given information about local and absolute maximums and minimums.
• Be able to find the critical numbers of a function.
• Be able to determine local and absolute maximums and minimums for a function on a closed interval using critical numbers and endpoints.
4.2 What Derivatives Tell Us
• Be able to find the intervals where a function is increasing and decreasing using a graph.
• Be able to find the intervals where a function is concave up and concave down using a graph.
• Be able to find the intervals where a function is increasing, decreasing, concave up and concave down given its formula using derivatives.
• Be able to find the local maximums and minimums for a function using the First Derivative Test or the Second Derivative Test.
• Be able to find the inflection points of a function using the second derivative.
• Be able to sketch the graph a function using information about its first and second derivatives.
• Be able to sketch the graph of the derivative of a function given the graph of the function.
• Be able to sketch the graph of a function given the graph of the derivative of the function.
4.3 Graphing Functions (Extra Credit)
• Be able to graph a function using information about its domain, x – intercept(s), y – intercept, horizontal asymptote(s), vertical asymptote(s), intervals of increasing and decreasing, local maximums and minimums, concavity and points of inflection.
4.4 Optimization Problems
• Be able to find the objective function and the constraint(s) needed to solve an applied problem.
• Be able to solve a wide variety of applied problems involving optimization.
4.5 Linear Approximations and Differentials
• Be able to find the linear approximation (Linearization) of a function at a value of x = a.
• Be able to calculate the differential of a function.
• Be able to find the linear approximation of a function at a value of x = a.
• Be able to calculate both dy and for a given function with a known value of x = a and a particular dx.
• Be able to approximate a function value using either a linear approximation or differential.
4.6 Mean Value Theorem (Extra Credit)
• Be able to determine when Rolle’s Theorem can be applied to a function and be able to find the value(s) of x = c that is/are guaranteed by the theorem.
• Be able to determine when the Mean Value Theorem can be applied to a function and be able to find the value(s) of x = c that is/are guaranteed by the theorem.
4.7 L’Hopital’s Rule (Extra Credit)
• Be able to determine whether a limit is of indeterminate form.
• Be able to change a limit into an indeterminate form, if possible.
• Be able to use L’Hopital’s Rule to evaluate limits that are of indeterminate form.
Chapter 3 Review (p. 232) 12 – 14, 17, 19, 23 – 31 odd, 37, 39, 41, 43, 49, 51, 67, 69 - 73