for smooth progress toward readiness for further study/careers in science, technology, engineering, and mathematics (STEM) fields. These fluencies are highlighted to stress the need to provide sufficient supports and opportunities for practice to help students gain fluency. Fluency is not meant to come at the expense of conceptual understanding. Rather, it should be an outcome resulting from a progression of learning and thoughtful practice. It is important to provide the conceptual building blocks that develop understanding along with skill toward developing fluency. References: TN State Standards Content Instructional Support & Resources Chapter 3: Applications of Differentiation (Allow approximately 3 weeks for instruction, review, and assessment) Domain: Computing and Applying Derivatives Cluster: Apply differentiation techniques Use first and second derivatives to analyze a function 7. Relate the increasing and decreasing behavior of f to the sign of f’ both analytically and graphically. 8. Use the derivative to find extrema (local and global). 9. Analytically locate the intervals on which a function is increasing, decreasing, or neither. 10. Relate the concavity of f to the sign of f” both analytically and graphically. 11. Use the second derivative to find points of inflection as points where concavity changes. Enduring Understandings: The derivative has both theoretical and real life applications. The derivative provides useful information the behavior of functions and the shapes of graphs. Understanding the rate of change of a function allows you to predict future behavior. Essential Questions: What does the derivative tell us? How can the derivative be used to solve optimization problems? How do rates of change relate in real-life settings? Objectives: Students will: 3.6: A Summary of Curve Sketching Additional Resource(s) Larson Calculus Videos – Section 3.6 s Chapter 3 Vocabulary (3-6, 3-7 & asymptotes, critical numbers, points of inflection, test intervals, primary equation, secondary equation, differential of x, differential of y, propagated error, differential form Writing in Math/Discussion Suppose f” (t) < 0 for all t in the interval (2, 8). Explain why f (3) > f (5).
12. Analytically locate intervals on which a function is concave up or concave down. Analyze the graphs of polynomials, rational, radical, piecewise, and transcendental functions using appropriate technology. Discuss which functions behave “nicely” with respect to algebraic properties and which do not. Justify your discussions. Describe asymptotic behavior (analytically and graphically) in of infinite limits and limits at infinity. Given a complete set of algebraic information and calculus information, construct a sketch of a function that matches the given data. Display several functions that satisfy one set of data. Describe their differences and similarities. Give a sketch of a of a function and completely describe the function in mathematical terms so that the sketch could be replicated from the description and would be close to the original graph Domain: Computing and Applying Derivatives Cluster: Apply derivatives to solve problems 16. Solve optimization problems to find a desired maximum or minimum value. Objectives: Students will: Use optimization to find extreme values (relative and absolute). 3.7: Optimization Problems Additional Resource(s) Larson Calculus Videos – Section 3.7 s Writing in Math/Discussion A shampoo bottle is a right circular cylinder. Because the surface area of the bottle does not change when it is squeezed, is it true that the volume remains the same? Explain. Domain: Computing and Applying Derivatives Cluster: Apply derivatives to solve problems 18. Use tangent lines to approximate function Objectives: Students will: Understand the concept of a tangent line approximation. 3.9: Differentials Additional Resource(s) Larson Calculus Videos – Section 3.9 Writing in Math/Discussion Describe the change in accuracy of dy as an approximation for Δy when Δx is decreased.
Fundamental Theorem of Calculus. Use the Fundamental Theorem of Calculus to represent a particular antiderivative of a function and to understand when the antiderivative so represented is continuous and differentiable. Apply basic properties of definite integrals. Understand and use the Mean Value Theorem for integrals. Find the average value of a function over a closed interval. Understand and use the Second Fundamental Theorem of Calculus. Khan Academy Fundamental Theorem of Calculus Videos Domain: Calculate and Apply Integrals Cluster: Apply techniques of antidifferentiation Develop facility with finding antiderivatives that follow directly from derivatives of functions (power, exponential, logarithmic, and trigonometric). Use substitution of variables to calculate antiderivatives (including changing limits for definite integrals). Find specific antiderivatives using initial conditions. Objectives: Students will: Use pattern recognition to find an indefinite integral. Use change of variables to find an indefinite integral. Use the general power rule for integration to find an indefinite integral. Use a change of variables to evaluate a definite integral. Evaluate a definite integral involving an even or odd function. 4.5: Integration by Substitution Additional Resource(s) Larson Calculus Videos – Section 4.5 Khan Academy Calculus Videos Domain: Understanding Integrals Cluster: Demonstrate understanding of a Definite Integral 3. Use Riemann sums (left, and midpoint) and trapezoidal sums to approximate definite integrals of functions, represented graphically, numerically, and by tables of values. Objectives: Students will: Approximate a definite integral using the Trapezoidal Rule. 4.6: Numerical Integration Additional Resource(s) Larson Calculus Videos – Section 4.6 Khan Academy Calculus Videos