Day 50 - Area Under a Curve - 10.28.14

Updates
  • Unit 4 Test will be on Friday, 11/14!
Bell Ringer
  • Antiderivative

  1. Find the following

    1. 5x2 + 5x + C

    2. 10x2 + 5x + C

    3. 5x2 + 5x

    4. 10

    5. none of the above

  2. Find the following

    1. 4cos(x) - 3sin(x) + C

    2. -4cos(x) + 3sin(x)

    3. -4cos(x) - 3sin(x) + C

    4. -4cos(x) + 3sin(x) + C

    5. -3cos(x) + 4sin(x) + C

  3. Find the following

    1. 4/t - 3/t2 + 2/t3 + C

    2. -4/t - 3/t2 + 2/t3 + C

    3. -4/t + 3/t2 + 2/t3

    4. -4/t - 3/t2 + 2/t3

    5. -4/t + 6/t2 + 6/t3 + C

  4. Find F(x) if F’(x) = -6x2 - 3 given F(2) = -3

    1. F(x) = -2x3 + 19

    2. F(x) = -2x3 + C

    3. F(x) = -12x3 - 3x + 19

    4. F(x) = -2x3 - 3x + 19

    5. F(x) = -2x3 + 3x

Review
  • Prerequisites
    • Derivatives
      • Derivative Rules
        • Power Rule
        • Constant Rule
      • 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