Download Math 217 Midterm Review Sheet: Useful Formulas and Identities and more Study notes Advanced Calculus in PDF only on Docsity! CRIB SHEET FOR MATH 217 MIDTERM This sheet will be handed out stapled to the midterm. You may freely use the following formulae and facts. Some of these will not be needed, and some formulae and facts will be needed that are not given here - those will be more standard and things you are expected to know already. Acceleration due to gravity is 32 feet per second squared. The area of a circle of radius r is πr2; its circumference is 2πr. The area of a triangle with base b and height h is 12bh. The volume of a cylinder of height h and radius r is πr2h. The volume of a cone of height h and radius r is 13πr 2h. The volume of a sphere of radius r is 43πr 3; its surface area is 4πr2. Pythagoras’s theorem: a2 + b2 = c2, where a, b, c are the sides of a right triangle with c the hypotenuse. The circle with center the origin and radius r has equation x2 + y2 = r2. tan(t) = sin(t)cos(t) , cot(t) = cos(t) sin(t) , sec(t) = 1 cos(t) , csc(t) = 1 sin(t) . sin(2t) = 2sin(t)cos(t), cos(2t) = cos2t − sin2t, cos2t + sin2t = 1, sin( t2 ) = ± √ 1−cos(t) 2 , cos( t 2 ) = ± √ 1+cos(t) 2 . sin(0) = 0 = cos(π2 ), cos(0) = 1 = sin( π 2 ), sin( π 6 ) = 1 2 = cos( π 3 ), sin( π 4 ) = √ 2 2 = cos(π4 ), sin( π 3 ) = √ 3 2 = cos( π 6 ). If a 6= 0, then the roots of ax2 + bx+ c = 0 are x = −b± √ b2−4ac 2a . aman = am+n, a m an = a m−n, (am)n = amn, a−n = 1/an. d dx (sin(x)) = cos(x), d dx (cos(x)) = −sin(x), d dx (tan(x)) = sec 2(x), ddx (cot(x)) = −csc2(x), ddx (sec(x)) = sec(x)tan(x), d dx (csc(x)) = −csc(x)cot(x). sinh(x) = (ex − e−x)/2, cosh(x) = (ex + e−x)/2. d dx (sinh(x)) = cosh(x), d dx (cosh(x)) = sinh(x), d dx (tanh(x)) = sech 2(x), ddx (coth(x)) = −csch2(x), ddx (sech(x)) = −sech(x)tanh(x), d dx (csch(x)) = −csch(x)coth(x). d dx (ln(x)) = 1/x, d dx (e x) = ex. d dx (sin −1(x)) = 1/ √ 1− x2, ddx (cos −1(x)) = −1/ √ 1− x2, ddx (tan −1(x)) = 1/(1+ x2), ddx (sec −1(x)) = 1/(|x| √ x2 − 1). d dx (sinh −1(x)) = 1/sqrtx2 + 1, ddx (cosh −1(x)) = 1/sqrtx2 − 1, ddx (tanh −1(x)) = 1/(1− x2), ddx (sech −1(x)) = −1/(x √ 1− x2).∫ audu = au/(ln(a)) + C, ∫ 1/udu = ln|u|+ C.∫ sin(u)du = −cos(u) + C, ∫ cos(u)du = sin(u) + C, ∫ sec2(u)du = tan(u) + C, ∫ csc2(u)du = −cot(u) + C, ∫ sec(u)tan(u)du = sec(u) + C, ∫ csc(u)cot(u)du = −csc(u)+C, ∫ tan(u)du = −ln|cos(u)|+C, ∫ cot(u)du = ln|sin(u)|+C, ∫ sec(u)du = ln|sec(u) + tan(u)|+ C, ∫ csc(u)du = ln|csc(u)− cot(u)|+ C.∫ 1/ √ a2 − u2du = sin−1(u/a)+C, ∫ 1/(a2+u2)du = (1/a)tan−1(u/a)+C, ∫ 1/(a2− u2)du = (1/2a)ln|(u+a)/(u−a)|+C, ∫ 1/(u √ u2 − a2)du = (1/a)sec−1(|u/a|)+C. Typeset by AMS-TEX 1
