Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) RECTANGULAR APPROXIMATION METHODS
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1 AP Calculus 5. Areas and Distances Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) Exercise : Calculate the area between the x-axis and the graph of y = 3 2x. Exercise 2: Calculate the area between the x-axis and the graph of y = 8 x 2 RECTANGULAR APPROXIMATION METHODS Exercise 3: Use rectangles to estimate the area under the parabola y = x 2 from to. Solution : Using RRAM with 4 subintervals Solution 2: Using LRAM with 4 subintervals Solution 3: Using MRAM with 4 subintervals
2 Exercise 4: Use rectangles to estimate the area under the curve y = + x 3 from -2 to 3. Solution : Using RRAM with 5 subintervals Solution 2: Using LRAM with 5 subintervals Solution 3: Using MRAM with 5 subintervals Exercise 5: Use rectangles to estimate the area under the curve y = x from to 6. Solution : Using RRAM with 2 subintervals Solution 2: Using LRAM with 2 subintervals Solution 3: Using MRAM with 2 subintervals
3 Definition: The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: A = lim n R n = lim n [f(x ) x + f(x 2 ) x + f(x 3 ) x + + f(x n ) x] Using sigma notation A = lim f(x i ) x n n i= Exercise 6: Let A be the area of the region that lies under the graph of f(x) = cos x between x = and x = π 2. Solution : Using RRAM with 4 subintervals Solution 2: Using LRAM with 4 subintervals Solution 3: Using MRAM with 4 subintervals
4 THE DISTANCE PROBLEM Suppose you have the following problem to solve: A person drives their car at a rate of 5 mph for 3 hours straight. Find the distance traveled. Problems like the one above are fairly easy to solve, but do not model the real-world. What about the time where the person had to get up to speed or slow down? If the velocity varies, it s not as easy to find the distance traveled. Example 5: Suppose the odometer in your car is broken and you want to estimate the distance driven over a 3 second time interval. You take speedometer readings every 5 seconds and record them in the following table. Time (s) Velocity (mph) Draw a graph of the data. Solution : Using RRAM Solution 2: Using LRAM How could you obtain a more accurate calculation without fixing your odometer?
5 AP Calculus 5.2 Riemann Sums and The Definite Integral Goal: Calculate the exact area under a curve using Riemann Sums. Exercise : Calculate the total distance traveled given the velocity equation. v(t) = t 2 2t 8 from t 6 Exercise 2: Calculate the total distance traveled given the acceleration equation. a(t) = t + 4 v() = 5 and t Exercise 3: Calculate the total area under the curve of the f(x) = x [,8]
6 Exercise 4: Calculate the total area under the curve of the f(x) = x 2 [,2] Exercise 5: Calculate the total area under the curve of the f(x) = 3 + x 2 [,4]
7 AP Calculus 5.3 The Fundamental Theorem of Calculus MENTAL OBJECTIVE: Discover & understand the relationship between net-area and antiderivatives. PHYSICAL OBJECTIVE: Perform Riemann sum and integral calculations by hand and with a calculator. PART : Let s say you are walking at a certain speed. I m not going to tell you the speed, because it s changing, but I am going to give you the position function: Position function in feet: s(t) = t 2 + 5, t is in seconds Question: Find the displacement traveled during the first 5 seconds Time Start End Position (placement) Displacement: PART 2: Compute the velocity function given s(t) from PART. Complete the following chart to find the total displacement during the first 5 seconds. t v(t) Calculate to distance using RRAM, LRAM, and find the exact by evaluating the integral. RRAM LRAM EXACT Question: There is an easier, more unbelievable, seemingly improbable way to calculate the displacement rather than finding the area under the velocity curve. How can you calculate the displacement, using an integral?
8 AP Calculus Total Distance vs. Displacement Practice Given velocity or acceleration Function v(t) = Position function Displacement On Time Interval Total Distance On time interval The acceleration function in (m/s 2 ) and the initial velocity are given for a particle moving along a line. a(t) = 2t 6, v() = 5, t 6 a) Calculate the displacement of the particle after the 5 seconds. b) Calculate the distance traveled by the particle during the 5 seconds.
9 The Fundamental Theorem of Calculus, Part (FTC ) If f is continuous on [a, b], then the function g defined by g(x) = f(t)dt a x a x b Is continuous on [a, b] and differentiable on (a, b), and g (x) = f(x). Examples: This says that the derivative and integral are opposite operations. The Fundamental Theorem of Calculus, Part 2 (FTC 2) If f is continuous on [a, b], then b f(x)dx = F(b) F(a) a Where F Is any antiderivative of f, that is, a function F = f. Examples: This tells you how to evaluate an integral. #) #2)
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11 Homework 5.3: Calculate the derivative of the function g(x) g (x) x g(x) = t 3 + dt x g(x) = (2 + t 4 ) 5 dt x g(x) = t 2 sin t dt 2 r g(r) = x dx π g(x) = + sec t dt x g(x) = sin 3 t dt x 2 g(x) = cos t dt x x g(x) = sin 4 t dt 2 x 2 g(x) = + r 3 dr tan x g(x) = t + tdt cos x g(x) = ( + v 2 ) dv
12 HW: 5.3 Evaluate the integral with and without a calculator. Integral 2 (x 3 2x)dx Work and answer without a calculator Answer with a calculator. Round to 3 decimals. 5 6 dx 2 4 (5 2t + 3t 2 ) dt ( + 2 x4 2 5 x9 ) dx x 4 5dx 8 3 xdx 2 3 t 4 dt 2π cos θ dθ π 2 x(2 + x 5 ) dx (3 + x x)dx 9 x x dx 2 (y )(2y + )dy π/4 sec 2 t dt
13 AP Calculus 5.4 Indefinite Integrals & Net Change
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15 AP Calculus 5.5 The u -Substitution Rule
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