Calculus 2 Cheat Sheet PDF

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Good study tool to use in trying to study for calculus two....

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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 ´ ( A , B , C ) and a point P(x 0 , y 0 , z 0) the equation for the plane is: you have N

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 ( sin θ ) Between points and planes, two parallel planes, lines and planes, or skew lines you use the dot product ( cos θ ), ( A´

compB´ ) 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

b

b

a

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:

1 2 2 W =∆ KE= m(v 2 −v 1 ) , make sure the m in the equation is in SLUGS for U.S. units 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 , where h is depth and w is weight-density, if the object is not at b

an angle it is dF = whLdh ⇒F=∫ 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, ´h 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 du sin−1 u= dx √ 1−u2 dx

……which is also equal to the negative

d cos−1 u dx 1 du d tan −1 u= ……..which is also equal to the negative dx 1+u2 dx d cot−1 u dx du d 1 −1 sec u= 2 dx |u|√ u −1 dx

……which is also equal to the negative

d csc−1 u dx

Integrals resulting from the derivatives:

∫

du =sin−1 u+C √1−u2

du

∫ 1+u2 =tan−1 u+C ¿ u∨¿ +C ∫ du2 =sec−1 ¿ 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 (e x):

dy dx

Integral of the natural exponential (e x):

∫ eu du=eu + C NOTE:

ln x

e =x

…… and ….. loga u=

Derivative of other exponentials (au):

ln u ln a

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 sinh x=

e x −e− x 2

cosh x =

e x + e−x 2

tanh x=

sinh x h

x

−x

e x −e− x = e +e

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

∫ x2 dx

would turn into

−∞

lim a →−∞

∫a

x 2 dx

For integrals that violate FTC you use the same idea 5 −¿

a→0

a

5



Example:

∫ dx 2 −3 x

dx

∫ x2

would have a divide by zero so you split it up into

b

b→0

+¿

+ lim ¿ ∫ dx 2 ¿ x

−3

lim ¿ ¿

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...