Calculus Unit 7

Big Idea:

Integrals and The Fundamental Theorem of Calculus

Student Expectations:

Focus Standards

EU 3.1: Antidifferentiation is the inverse process of differentiation.

• EK 3.1A1: An antidervative of a functionis a function whose derivative is .
• EK 3.1A2: Differentiation rules provide the foundation for finding antiderivatives.

EU 3.2: The definite integral of a function over an interval is the limit of a Riemann sum over that interval and can be calculated using a variety of strategies.

• EK 3.2A1: A Riemann sum, which requires the partition of an interval , is the sum of products, each of which is the value of a function at a point in the subinterval multiplied by the length of that subinterval of the partition.
• EK 3.2A2: The definite integral of a continuous function over the interval denoted by , is the limit of Riemann sums as the widths of subintervals approach That is, where is a value in the i^th subinterval, is the width of the i^th subinterval, n is the number of subintervals, and is the width of the largest subinterval. Another form of the definition is , where and is a value in the i^th subinterval.
• EK 3.2A3: The information in the definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral.
• EK 3.2B1: Definite integrals can be approximated for functions that ar represented graphically, numerically, algebraically, and verbally.
• EK 3.2B2: Definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or non-uniform Partitions.
• EK 3.2C1: In some cases, a definite integral can be evaluated by using geometry and the connection between the definite integral and area.
• EK 3.2C2: Properties of definite integrals include the integral of a constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a function over adjacent intervals.
• EK 3.2C3: The definition of the definite integral may be extended to functions with removable or jump Discontinuities.

EU 3.3: The Fundamental Theorem of Calculus, which has two distinct formulations, connects differentiation and int3A1:egration.

• EK 3.3A1: The definite integral can be used to define new functions; for example, .
• EK 3.3A2: If is a continuous function on the interval , then where x is between and .
• EK 3.3A3: Graphical, numerical, algebraic, and verbal representations of a function provide information about the function defined as .
• EK 3.3B1: The function defined by is an antiderivative of .
• EK 3.3B2: If is a continuous function on the interval and is an antiderivative of , then .
• EK 3.3B3: The notation means that and is called an indefinite integral of the function .
• EK 3.3B4: Many functions do not have closed form antiderivatives.
• EK 3.3B5: Techniques for finding antiderivatives include algebraic manipulation such a long division and completing the square, substitution of variables, (BC) integration by parts, and nonrepeating linear partial fractions.

EU 3.4: The definite integral of a function over an interval is a mathematical tool with many interpretations and applications involving accumulation.

• EK 3.4A3: The limit of an approximating Riemann sum can be interpreted as a definite integral.

Student Learning Targets:

• I will recognize antiderivatives of basic functions
• I will interpret the definite integral as the limit of a Riemann sum
• I will express the limit of a Riemann sum in integral notation
• I will approximate a definite integral
• I will calculate a definite integral using areas and properties of definite integrals
• I will analyze functions defined by an integral
• I will calculate antiderivatives
• I will evaluate definite integrals

Essential Questions:

Extra Information:

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