After completing this section, students should be able to do the following.Define linear approximation as an application of the tangent to a curve.Find the linear approximation to a function at a point and use it to approximate the function value.Identify when a linear approximation can be used.Label a graph with the appropriate quantities used in linear approximation.Find the error of a linear approximation.Compute differentials.Use the second derivative to discuss whether the linear approximation over or underestimates the actual function value.Contrast the notation and meaning of dydy versus ΔyΔy.Understand that the error shrinks faster than the displacement in the input.Justify the chain rule via the composition of linear approximations.

After completing this section, students should be able to do the following.Identify situations where logs can be used to help find derivatives.Use logarithmic differentiation to simplify taking derivatives.Take derivatives of functions raised to functions.Recognize the difference between a variable in the base and a variable in the exponent.

After completing this section, students should be able to do the following.Define a critical point.Find critical points.Define local maximum and local minimum.Classify critical points.State the First Derivative Test.Apply the First Derivative Test.Define inflection points.Find inflection points.State the Second Derivative Test.Apply the Second Derivative Test.

After completing this section, students should be able to do the following.Understand the statement of the Extreme Value Theorem.Understand the statement of the Mean Value Theorem.Sketch pictures to illustrate why the Mean Value Theorem is true.Determine whether Rolle’s Theorem or the Mean Value Theorem can be applied.Find the values guaranteed by Rolle’s Theorem or the Mean Value Theorem.Use the Mean Value Theorem to solve word problems.Compare and contrast the Intermediate Value Theorem, Mean Value Theorem, and Rolle’s Theorem.Identify calculus ideas which are consequences of the Mean Value Theorem.

After completing this section, students should be able to do the following.Solve basic related rates word problems.Understand the process of solving related rates problems.Calculate derivatives of expressions with multiple variables implicitly.

After completing this section, students should be able to do the following.Identify products of functions.Use the product rule to calculate derivatives.Identify quotients of functions.Use the quotient rule to calculate derivatives.Combine derivative rules to take derivatives of more complicated functions.Explain the signs of the terms in the numerator of the quotient rule.Use the product and quotient rule to calculate derivatives from a table of values.

After completing this section, students should be able to do the following.Know the graphs and properties of ‘‘famous’’ functions.Know and use the properties of exponential and logarithmic functions.Understand the relationship between exponential and logarithmic functions.Understand the definition of a rational function.Understand the properties of trigonometric functions.Evaluate expressions and solve equations involving trigonometric functions and inverse trigonometric functions.

After completing this section, students should be able to do the following.Use the definition of the derivative to develop shortcut rules to find the derivatives of constants and constant multiples.Use the definition of the derivative to develop shortcut rules to find the derivatives of powers of xx.Use the definition of the derivative to develop shortcut rules to find the derivatives of sums and differences of functions.Compute the derivative of polynomials.Recognize different notation for the derivative.State the derivative of the natural exponential function.State the derivative of the sine function.

After completing this section, students should be able to do the following.State the Second Fundamental Theorem of Calculus.Evaluate definite integrals using the Second Fundamental Theorem of Calculus.Understand how the area under a curve is related to the antiderivative.Understand the relationship between indefinite and definite integrals.

After completing this section, students should be able to do the following.Undo the chain rule.Calculate indefinite integrals (antiderivatives) using basic substitution.Calculate definite integrals using basic substitution.