Make an AP calculus class for a grade 11 student. Make sure it covers most of AP calculus curriculum.

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Welcome to AP Calculus, a college-level mathematics course designed for 11th grade students. This curriculum covers the fundamental concepts of calculus including limits and continuity, differentiation, integration, differential equations, and for BC calculus, infinite series and sequences. The course builds on your precalculus knowledge and develops both conceptual understanding and problem-solving skills needed for the AP exam. Welcome to AP Calculus. This course covers the fundamental concepts of calculus that are essential for advanced mathematics and many scientific fields. AP Calculus is divided into two main branches: differential calculus, which deals with rates of change, and integral calculus, which focuses on the accumulation of quantities. Throughout this course, we'll explore five main topics: limits and continuity, derivatives, applications of derivatives, integrals, and applications of integrals. Each topic builds upon the previous ones, creating a cohesive mathematical framework for understanding change and accumulation. Let's begin with limits and continuity, the foundation of calculus. A limit describes the value a function approaches as the input approaches a particular value. We write this as 'the limit of f(x) as x approaches a equals L.' For example, in this graph, as x approaches 2 from either side, the function approaches the value 2, so we say the limit equals 2. A function is continuous at a point if the function is defined at that point, the limit exists, and the limit equals the function value. In our example, there's a hole at x equals 2, making the function discontinuous at that point. Derivatives represent the rate of change of a function at a specific point. Formally, the derivative is defined as the limit of the difference quotient as h approaches zero. This formula gives us the instantaneous rate of change. There are several notations for derivatives, including f prime of x, df dx, and dy dx. In this graph, we see the function f of x equals x squared. At the point x equals 2, the derivative equals 4, which is the slope of the tangent line at that point. This tells us the rate at which the function is increasing at x equals 2. Derivatives have many important applications in calculus. One key application is in motion, where the derivative represents velocity and the second derivative represents acceleration. Optimization problems involve finding maximum and minimum values, which occur at critical points where the derivative equals zero. In this graph, we can see a function with a minimum and a maximum point. Related rates problems explore how different rates of change are connected through the chain rule. Linear approximation uses the tangent line to approximate function values near a point where the derivative is known. These applications make calculus a powerful tool for solving real-world problems. Integrals represent the accumulation of quantities and are often visualized as the area under a curve. The definite integral from a to b of f of x dx gives us the net signed area between the function and the x-axis over the interval from a to b. The Fundamental Theorem of Calculus establishes the profound connection between derivatives and integrals, stating that the integral of a function's derivative equals the total change in the original function. This revolutionary insight shows that differentiation and integration are inverse processes. Applications of integrals include calculating areas, volumes, work, energy, average values, and probabilities, making integration a powerful tool across mathematics, physics, engineering, and other disciplines. Derivatives represent the rate of change of a function at a specific point. Formally, the derivative is defined as the limit of the difference quotient as h approaches zero. This formula gives us the instantaneous rate of change. There are several notations for derivatives, including f prime of x, df dx, and dy dx. In this graph, we see the function f of x equals x squared. At the point x equals 2, the derivative equals 4, which is the slope of the tangent line at that point. This tells us the rate at which the function is increasing at x equals 2. Derivatives have many important applications in calculus. One key application is in motion, where the derivative represents velocity and the second derivative represents acceleration. Optimization problems involve finding maximum and minimum values, which occur at critical points where the derivative equals zero. In this graph, we can see a function with a minimum and a maximum point. Related rates problems explore how different rates of change are connected through the chain rule. Linear approximation uses the tangent line to approximate function values near a point where the derivative is known. These applications make calculus a powerful tool for solving real-world problems. Integrals represent the accumulation of quantities and are often visualized as the area under a curve. The definite integral from a to b of f of x dx gives us the net signed area between the function and the x-axis over the interval from a to b. The Fundamental Theorem of Calculus establishes the profound connection between derivatives and integrals, stating that the integral of a function's derivative equals the total change in the original function. This revolutionary insight shows that differentiation and integration are inverse processes. Applications of integrals include calculating areas, volumes, work, energy, average values, and probabilities, making integration a powerful tool across mathematics, physics, engineering, and other disciplines.