Update
Questions
Bell Ringer
Review
- Prerequisites
- Derivatives
- Derivative Rules
- Power Rule
- Constant Rule
- Trig Derivatives
- Antiderivative (video)
- Indefinite Integral
- Definition
- Integral Sign
- Integrand
- Variable of Integration
- Constant of Integration
- General Solution
- Basic Integration Rules
- Summation (Sigma Notation) (video)
- Area Under a Curve (video 1) (video 2) (video 3)
- Fundamental Theorem of Calculus
- Definite Integrals
- Integration by Substitution (video 1) (video 2) (video 3)
- How can integration by substitution be used to solve problems?
- Integrate by substitution.
- Checkpoints
Lesson
Exit Ticket
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Lesson Objective(s)
Skills
- Find critical numbers using differentiation.
- Find extrema on a closed interval using differentiation.
- Apply understanding of Rolle’s Theorem and the Mean Value Theorem.
- Determine intervals on which a function is increasing or decreasing.
- Apply the First Derivative Test to find relative extrema of a function.
- Determine intervals on which a function in concave upward or downward.
- Apply the Second Derivative Test to find inflection points of a function.
- Determine horizontal asymptotes using limits.
- Determine finite limits at infinity.
- Determine infinite limits at infinity.
- Determine the conditions that optimize a situation.
- Calculate the antiderivative of a function.
- Calculating the area under a curve using summation.
- Calculate the area under a curve using definite integration.
- Integrate by substitution.
Standard(s)
- APC.10
- Use Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented algebraically, graphically, and by a table of values and will interpret the definite integral as the accumulated rate of change of a quantity over an interval interpreted as the change of the quantity over the interval
- Riemann sums will use left, right, and midpoint evaluation points over equal subdivisions.
- APC.11
- The student will find antiderivatives directly from derivatives of basic functions and by substitution of variables (including change of limits for definite integrals).
- APC.12
- The student will identify the properties of the definite integral. This will include additivity and linearity, the definite integral as an area, and the definite integral as a limit of a sum as well as the fundamental theorem.
- APC.13
- The student will use the Fundamental Theorem of Calculus to evaluate definite integrals, represent a particular antiderivative, and facilitate the analytical and graphical analysis of functions so defined.
- APC.14
- The student will find specific antiderivatives, using initial conditions (including applications to motion along a line). Separable differential equations will be solved and used in modeling (in particular, the equation y' = ky and exponential growth).
- APC.15
- The student will use integration techniques and appropriate integrals to model physical, biological, and economic situations. The emphasis will be on using the integral of a rate of change to give accumulated change or on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. Specific applications will include
- a) the area of a region;
- b) the volume of a solid with known cross-section;
- c) the average value of a function; and
- d) the distance traveled by a particle along a line.
Past Checkpoints - Antiderivative
- Area of a Plane Region
- Fundamental Theorem of Calculus
- Definite Integrals
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