alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# 1 area of a trapezoid is s um of the parallel sides

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: must be a polynomial o f the form 1 ax 2 + b x + C polynomial pol -2 For : dy(t) = -- g t + u.si n θ dt 1 y(t) = ----- g t 2 + u .si n θ . t + may be some constant C. 2 if at time t = 0 the ball was thrown from the ground level ( = 0) : C = 0 if at time t = 0 the ball was thrown from a height of 5 meters: C = + 5m This mechanical method of calculation wor ks when the functions are INTEGRABLE. It is known as finding the ANTIDERIVATIVE. There are tables which you can look up to find the matching ANTIDERIVATIVE of the given DERIVATIVE. Analyticall ytically Analytically, if f(x) is the DERIVATIVE of the function F(x), then the function F(x) is called the ANTIDERIVATIVE of the function f(x). We may write: d F(x) f(x) = dx If we write f(x)dx = dF(x) then we call f(x) the DIFFERENTIAL COEFFICIENT. 129 Integ Below is a partial Table of Integr als of the more frequently encountered functions deriv tiv integ antideriv tiv integ in standard form. The derivative = integr and and antiderivative = integr al . deri antideri DERIVA ANTIDERIVA F(x) DERIVATIVE f(x) ANTIDERIVATIVE F(x) n+1 x xn n ≠ −1 n+1 1 log |x| /x ex ex ax x a a > 0, a ≠ 1 log a sin (x) − cos x cos (x) sin (x) sec2 (x) tan (x) cosec2 (x) − cot (x) tan (x) . sec (x) sec (x) cot (x) . cosec (x) − cosec (x) tan (x) log |sec (x)| cot (x) log |sin (x)| sec (x) log|sec (x) + tan (x)|= log|tan (π 4 + x/ 2)| / cosec (x) log|cosec (x) − cot (x)|or log|tan x/ 2| 1 a + x2 1 2 a − x2 1 tan −1 ( x ) a ≠ 0 /a /a a+x 1 --- log | | 2a a−x 1 x − a2 1 x−a --- log | | 2a x+a 2 2 1 √a2 − x2 x sin −1(|a|) x2 < a2 130 26. 26. F(x) = area under f(x) = ∫ f(x)dx f(x)dx Let us look at the original concept of the integral - finding the area under a integral area curv cur ve . For the time being we denote the area function as F(x). area curv F(x) = ar ea under the cur v e of f(x) In the next chapter we shall prove that : {f(x) f(x)} F(x ANTIDERIVA f(x)dx F( x ) = ANTIDERIVATIVE {f(x)} = ∫ f(x)dx area We now present 2 methods to find...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern