Updates- Unit 4 Test will be open note!
- Unit 4 Test will be on Friday, 11.14.14
Bell RingerApproximate the area under the function f(x) = 2x + 5 on the closed interval [0, 2] using 4 rectangles. Upper Area = 13; Lower Area = 15 Upper Area = 13; Lower Area = 14 Upper Area = 12; Lower Area = 15 Upper Area = 10; Lower Area = 15 none of the above
Calculate the actual area under the function f(x) = 2x + 5 on the closed interval [0, 2]. 12 13 14 15 none of the above
Approximate the area under the function f(x) = cos(x) on the closed interval [0, π/2] using 4 rectangles. Upper Area = 1.1835; Lower Area = 0.7908 Upper Area = 0.1835; Lower Area = 1.7908 Upper Area = 2.1835; Lower Area = 0.7908 Upper Area = 1.1835; Lower Area = 1.7908 none of the above
Calculate the actual area under the function f(x) = cos(x) on the closed interval [0, π/2]. 0 1 2 π - none of the above
Review- Prerequisites
- Derivatives
- Derivative Rules
- Trig Derivatives
- Antiderivative
- Indefinite Integral
- Definition
- Integral Sign
- Integrand
- Variable of Integration
- Constant of Integration
- General Solution
- Basic Integration Rules
Lesson Exit Ticket- Posted on the board at end of the block.
| Lesson Objective(s)- How can the area under a curve be approximated?
- How can the area under a curve be calculated?
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.
Mathematical Practice(s)- #1 - Make sense of problems and persevere in solving them
- #2 - Reason abstractly and quantitatively
- #5 - Use appropriate tools strategically
- #6 - Attend to precision
- #8 - Look for and express regularity in repeated reasoning
Past Checkpoints |