In this course, students will apply limits to define definite integrals and how the Fundamental Theorem connects integration and differentiation. Students will also work to understand the theoretical basis and solve problems. By modifying the College Board's existing course structure, Scholar Sphere has optimized this course to make learning more efficient.
Subject Breakdown:
Unit 1: Limits and Continuity
Limits characterize how a function behaves as the input nears a specific value, irrespective of the function's actual value at that point. Continuity demands that the function's behavior near a point aligns with its value at that point. These fundamental concepts are crucial throughout calculus.
Unit 4: Introduction to Integrals and Their Applications
In this unit, students will learn about antiderivatives, definite integrals, and the Fundamental Theorem of Calculus. They will explore Riemann sums, sigma notation, and u-substitution. Practical applications include calculating definite integrals and finding the average value of functions.
Unit 7: Advanced Techniques in Area and Volume Calculations
Unit 7 explores advanced calculus topics focusing on area calculations using the integral and theorems such as the Intermediate Value Theorem (IVT), Extreme Value Theorem (EVT), and Squeeze Theorem. It also covers volume calculations using methods like the Disk Method and Washer Method for solids of revolution.
Unit 2: Differentiation Techniques and Applications
In this unit, students will learn key methods for finding derivatives, including basic rules, the product and quotient rules, and the chain rule. They will apply these techniques to exponential and logarithmic functions and explore practical uses such as calculating rates of change and using tangent lines for linear approximations. This unit provides a comprehensive understanding of differentiation and its real-world applications.
Unit 3: Advanced Differentiation and Applications
This unit covers advanced differentiation techniques like implicit differentiation and related rates. Students will analyze function behavior, including increasing/decreasing intervals, extreme values, concavity, and points of inflection. The unit also includes practical applications such as optimization problems.
Unit 5: Advanced Derivatives and Applications
This unit covers advanced topics in differentiation, including transcendental derivatives, logarithmic techniques, L'Hopital's Rule, and derivatives of inverse functions. It also revisits key concepts like implicit differentiation and related rates, with a focus on practical applications.
Unit 6: Advanced Integration Techniques and Applications
Unit 6 focuses on advanced integration techniques such as transcendental indefinite and definite integrals, inverse trigonometric derivatives and integrals, and applications of the Fundamental Theorem of Calculus with transcendental functions. It also covers topics like exponential growth and decay, preparing students for comprehensive understanding and application of integration concepts.
Unit 8: Exploring Differential Equations and Their Solutions
Unit 8 delves into the study of differential equations, covering topics such as slope fields, solutions to differential equations, and applications involving transcendental functions. Students will gain a deeper understanding of how differential equations model various real-world phenomena and explore methods to solve them effectively.
Extra Practice (practice tests)
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