Calculus 2 Cheat Sheet

Calculus 2 Cheat Sheet Unit One To get equation for a plane you need: a vector normal (perpendicular) to the plane and a point on the plane, for example if you have Ń ( A ,B,C ) and a point P( x 0 ,y 0 ,z 0) the equation for the plane is: A ( x − x 0 ) +B( y− y 0 ) +C ( z − z 0 ) =0 To get the parametric equations for a line you need: a vector parallel to the line and a point on the line, for example if you V ¿∨¿ (a , b , c) and a point ´¿ have x = x 0 +at , y= y 0 +bt , P( x 0 ,y 0 ,z 0) the equations for the line will be: z=z 0 +ct To find the distance between points, lines, and planes: Between points and lines or two parallel lines you use the cross product ( sinθ ) Between points and planes, two parallel planes, lines and planes, or skew lines you use the dot product ( cos θ ), ( Á compB́ ) Unit Two For a spring/rubber band: F=kx , where k is the spring constant and x is the displacement b W =∫ kxdx , where x is a variable and k is the spring constant, you can also solve for k if you have W a For a gas/piston cylinder: k PV =C P2 V 2− P1 V 1 1−k For leaking bucket/sand: b ∫ f ( x) dx , where f(x) is the function of the leaking bucket/sand a For conical pile/hemisphere pumping out water: b ∫ a b b a a hdF=∫ hwdV=∫ hwAdy , where h is the height (in terms of y), w is the weight-density, and A is the area Using work-kinetic energy relationship: W =∆KE= 1 m(v2 −v 2 ) , make sure the m in the equation is in SLUGS for U.S. units 2 1 2 Example for if mass is given in ounces: go from ounces to pounds, divide by 32 (U.S. gravitational constant) and plug into the equation Hydrostatic Force: HF = pA dF = pdA , , p= wh ⟹ dF =whdA dF = whLdh ⇒F=∫ an angle it is , where h is depth and w is weight-density, if the object is not at b whLdh , if it is at an angle it becomes dF = whLds , and you have to find dh in a terms of ds F=w ´h A , where w is weight-density, h́ is the centroid (which is 1/3h for a triangle), and A is the area Unit Three To find the inverse of a function, find x in terms of y and then switch the x and y  For example, the inverse of y = x 3 y=x Remember would be 1 3 −1 −1 f ( f ( x ))=f ( f ( x ) )=x Derivatives of the inverse trigonometric functions: d −1 sin u= dx du 1 √ 1−u2 dx ……which is also equal to the negative d cos−1 u dx 1 d du −1 tan u= ……..which is also equal to the negative 2 dx 1+u dx d cot−1 u dx d du 1 −1 sec u= dx |u| √ u2−1 dx ……which is also equal to the negative d csc−1 u dx Integrals resulting from the derivatives: du ∫ √1−u2 =sin −1 u+C ∫ du =tan−1 u+C 2 1+u ∫ ¿ u∨¿ +C du =sec−1 ¿ 2 u √u −1 DON’T FORGET ABSOLUTE VALUE FOR sec-1(x) AND csc-1(x) Unit Four Derivative of the natural log (ln): Common natural logs: Integral of the natural log (ln): u You can implicitly differentiate using ln is you take the natural log of both sides and solve for Derivative of the natural exponential (ex): dy dx Integral of the natural exponential (ex): ∫ e u du=eu + C NOTE: e lnx =x …… and ….. loga u= Derivative of other exponentials (au): ln u lna Integral of other exponentials (au): u(x)v(x): Derivative of Exponential Applications: solve for ek and then solve for the needed value For non-compound interest rates (example: 2%) the rate would equal k For compound interest rates: where k is the number times compounded per year and t is the number of years Newtons Law of Cooling: where T is the actual temperature, Ts is the temperature of the surroundings, and T0 is the initial temperature Unit Five x sinh x= −x e −e 2 coshx = e x + e−x 2 x tanh x= −x sinh x e −e = coshx e x +e− x Hyperbolic identitiy to remember: cosh2 x− sinh2 x=1 very similar to regular trigonemtric identites just remember that the derivative of cosh(x) is sinh(x) Catenary (chain): remember that, 1 =∞ 0 For integrals with one of the limits of integration being ∞ , turn it into a limit  10 Example: ∫ 10 x 2 dx would turn into −∞ 2 lim ∫ x dx a→−∞ a For integrals that violate FTC you use the same idea −¿ a→0  5 Example: ∫ −3 dx x2 would have a divide by zero so you split it up into b→0 +¿ ∫ 5 a dx 2 x b dx + lim ¿ 2 ¿ x lim ¿ ∫ −3 ¿ Unit Six Basic integration techniques: Forms of substitutions using the unit circle with a and x: Integration by parts: Partial Fractions:  Method One (equating coefficiants)  Method Two (heaviside method) Unit Seven Geometric series: Unit Eight THESE ARE THE SAME AS FROM UNIT FIVE For binomial series, use the form (1 + x )m Unit Nine