### Day 65 - Summative Exam 2 - 04.21.15

Update
• N/A

Questions

Bell Ringer
• N/A

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?
1. Integrate by substitution.
1. Checkpoints

Lesson
• Summative Exam 2

Exit Ticket
• N/A

Lesson Objective(s)
• Summative Exam 2
Skills
1. Find critical numbers using differentiation.
2. Find extrema on a closed interval using differentiation.
3. Apply understanding of Rolle’s Theorem and the Mean Value Theorem.
4. Determine intervals on which a function is increasing or decreasing.
5. Apply the First Derivative Test to find relative extrema of a function.
6. Determine intervals on which a function in concave upward or downward.
7. Apply the Second Derivative Test to find inflection points of a function.
8. Determine horizontal asymptotes using limits.
9. Determine finite limits at infinity.
10. Determine infinite limits at infinity.
11. Determine the conditions that optimize a situation.
12. Calculate the antiderivative of a function.
13. Calculating the area under a curve using summation.
14. Calculate the area under a curve using definite integration.
15. Integrate by substitution.

#### In-Class Help Requests

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