Riemann sum Study guides, Class notes & Summaries

These notes are in a pdf format. - colour coded - step-by-step examples for each topic The notes cover the topic: Antiderivatives, Mathematical Induction, Binomial theorem, Definite integral & Riemann sums

This document contains detailed in-depth notes about Riemann Sums (calculus) for grade 12 mathematics. The following topics are dealt with: 1.1 Simple Definition of Riemann Sums 1.2 Important Representations 1.3 Important Series Notes 1.4 Steps for Riemann Sums 1.5 Example 1 and Example 2 1.6 Additional Important Notes

The slope of a secant line represents what kind of change? - ANSWER-Average roc The slope of a tangent line represents what kind of change? - ANSWER-Instantaneous roc What kind of change does an integral represent? - ANSWER-Total change Explain what a derivative is graphically. - ANSWER-Slope of tangent line Explain what a derivative is numerically. - ANSWER-Difference quotient from a table Explain what a derivative is analytically. - ANSWER-lim of (f(x+h)-f(x))/h as h->0 3 ...

Integral of cosx - ANSWER-sinx+c Integral of sin x - ANSWER--cosx+c Integral of sec^2x - ANSWER-tanx+c Integral of secxtanx - ANSWER-secx+c Integral of csc^2x - ANSWER--cotx+c Integral of cscxcotx - ANSWER--cscx+c If a trapezoidal sum overapproximates ∫0 to 4 f(x)dx, and a right Riemann sum underapproximates ∫0 to 4 f(x)dx, which of the following could be the graph of y=f(x)? - ANSWER-A- The one with a y intercept at three that is concave up and decreasing Let f be the f...

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Math 141 In-Class Worksheet Week 5 Page 1 of 3 Question 1 Consider the triangle sliced below. (a) The area of the slice is approximately the area of a rectangle. Express the approximate slice area i n terms of hi and ∆x. (b) Use similar triangles to express the slice area in terms of the “slicing variables” or ”eventual integration variables” xi and ∆x. (c) Write a Riemann sum that approximates the total area of the triangle by adding up the areas of similar slices making up the enti...

SOLUTIONS MANUAL for Control Systems Engineering 7th Edition by Norman Nise. ISBN 9781118800638 Trapezoid = 31.516 Use the trapezoid rule with n = 4 to approximate the area between the curve f(x) = x^3 -x and the x-axis from x = 3 to x =7ANS-Trapezoid = (1/2)(1)[(3^3 -3) +2(4^3 -4) +2(5^3 -5) +2(6^3 -6) +(7^3 -7)] Trapezoid = 570 Use the trapezoid rule with n = 4 to approximate the area between the curve f(x) = x^2 + 1 and the x-axis from x = 3 to x =7ANSTrapezoid = (1/2)(1)[(3^2...

Integrals 1.1. Areas and Distances. The Definite Integral 1.1.1. The Area Problem. The Definite Integral. Here we try to find the area of the region S under the curve y = f (x) from a to b, where f is some continuous function. Y y=f(x) S O a b X In order to estimate that area we begin by dividing the interval [a, b] into n subintervals [x0 , x1 ], [x1 , x2 ], [x2 , x3 ], . . . , [xn−1 , xn ], each of length ∆x = (b − a)/n (so xi = a + i∆x). Y O a b 6 X 1.1. AREAS AND DISTANCES. THE DEFIN...

Defines Riemann sums, summations, and definite integrals with examples and diagrams

Calculus Notes-Questions and Answers Graded A+ tangent line approximation - ANSWER-y=f'(a)(x-a)+f(a) overestimate if f is concave down, underestimate if f is concave up f''(c) is positive - ANSWER-relative max f''(c) is negative - ANSWER-relative min f''(c)=0 - ANSWER-second derivative test fails; use first derivative test f''(x) is positive - ANSWER-concave up f'(x) increases f''(x) is negative - ANSWER-concave down f'(x) decreases directly proportional - AN...