Calculus Equations

Front 1

Mean Value Theorem for Derivatives

(MVTD)

Back 1

on the continuous & differentiable interval [a,b], there is at least one number, c, such that...

Hint 1

if y=f(x) is continuous on the interval [a,b] and is differentiable on the interval (a,b), then there is at least one number c between a and b such that...

Front 2

Rolles Theorum

Back 2

if, in a MVTD scenario,

f(a)=f(b)=0

then

f'(c)=0.

Hint 2

a version of MVTD

Front 3

d/dx * sinx = ?

Back 3

d/dx * sinx = cosx

Hint 3

cos?

Front 4

d/dx * cosx = ?

Back 4

d/dx * cosx = -sinx

Hint 4

sin?

Front 5

d/dx * tanx = ?

Back 5

d/dx * tanx = sec^2(x)

Hint 5

sec?

Front 6

d/dx * secx = ?

Back 6

d/dx * secx = secx*tanx

Hint 6

tan?

Front 7

d/dx * cscx = ?

Back 7

d/dx * cscx = -cscx * cotx

Hint 7

cot?

Front 8

d/dx * cotx = ?

Back 8

d/dx * cotx = -csc^2(x)

Hint 8

csc?

Front 9

tan(xy) = ?

Back 9

(tanx)(tany)

Hint 9

...

Front 10

sin(2x) = ?

Back 10

sin(2x) = 2*sinx*cosx

Hint 10

...

Front 11

cos(2x) = ?

Back 11

Hint 11

3 different unique expressions are equivalent to cos(2x)

Front 12

tan(2x) = ?

Back 12

Hint 12

its a fraction

Front 13

sin^2(x) = ?

Back 13

sin^2(x) = 1/2 - 1/2cos(2x)

Hint 13

cos?

Front 14

cos^2(x) = ?

Back 14

cos^2(x) = 1/2 + 1/2cos(2x)

Hint 14

cos?

Front 15

sinx - siny = ?

Back 15

Hint 15

involves fractions

Front 16

cosx - cosy = ?

Back 16

Hint 16

involves fractions

Front 17

sin^2(x) + cos^2(x) = ?

Back 17

sin^2(x) + cos^2(x) = 1

Hint 17

simple

Front 18

sec^2(x) = ?

Back 18

sec^2(x) = tan^2(x) + 1

Hint 18

tan?

Front 19

csc^2(x) = ?

Back 19

csc^2(x) = cot^2(x) + 1

Hint 19

cot?

Front 20

Back 20

Hint 20

cos?

Front 21

Pythagorean triangles/trig

Back 21

Hint 21

3 different trigonomentric equations based on Pythagorean trianges

Front 22

tanx = ?

Back 22

tanx = sinx/cosx

tanx = 1/cotx

Hint 22

sin?

Front 23

cotx = ?

Back 23

cotx = cosx/sinx

cotx = 1/tanx

Hint 23

cos?

Front 24

cscx = ?

Back 24

cscx = 1/sinx

Hint 24

sin?

Front 25

secx = ?

Back 25

secx = 1/cosx

Hint 25

cos?

Front 26

sinx = ?

Back 26

sinx = 1/cscx

Hint 26

csc?

Front 27

cosx = ?

Back 27

cosx = 1/secx

Hint 27

sec?

Front 28

if n is an integer...

sin(x+2(pi)n) = ?

Back 28

sin(x+2(pi)n) = sinx

Hint 28

its simple

Front 29

if n is an integer...

csc(x+2(pi)n) = ?

Back 29

csc(x+2(pi)n) = cscx

Hint 29

its simple

Front 30

if n is an integer...

cos(x+2(pi)n) = ?

Back 30

cos(x+2(pi)n) = cosx

Hint 30

its simple

Front 31

if n is an integer...

sec(x+2(pi)n) = ?

Back 31

sec(x+2(pi)n) = secx

Hint 31

its simple

Front 32

if n is an integer...

tan(x + (pi)n) = ?

Back 32

tan(x + (pi)n) = tanx

Hint 32

its simple

Front 33

if n is an integer...

cot(x + (pi)n) = ?

Back 33

cot(x + (pi)n) = cotx

Hint 33

its simple

Front 34

Back 34

Hint 34

its simple

Front 35

Back 35

Hint 35

its simple

Front 36

Back 36

Hint 36

its simple

Front 37

Back 37

Hint 37

its simple

Front 38

Back 38

Hint 38

its simple

Front 39

Back 39

Hint 39

its simple

Front 40

Derivitive Power Rule

Back 40

Hint 40

multiple, subtract one...

Front 41

Anti-Derivative Power Rule

Back 41

except when x=-1

Hint 41

...

Front 42

Back 42

Hint 42

anti-derivitive

Front 43

Back 43

Hint 43

anti-derivitive

Front 44

Back 44

Hint 44

anti-derivitive

Front 45

Back 45

Hint 45

anti-derivitive

Front 46

Back 46

Hint 46

anti-derivitive

Front 47

Back 47

Hint 47

anti-derivitive

Front 48

Back 48

Hint 48

separate

Front 49

Back 49

Hint 49

separate

Front 50

Average Value of a function

Back 50

Hint 50

expression

Front 51

Finding Critical Values

Back 51

if f'(c) = 0, the point C, (c,f(c)), is a critical value; setting the function to 0 and solving for x will give you all the Critical Values.

Hint 51

Point C test

Front 52

finding Points of Inflection

Back 52

With c being a critical value:

if f"(c) = 0, then the point C, (c,f(c)), is a point of inflection.

Hint 52

Point C test

Front 53

Finding Relative Maxs and Mins

Back 53

If a Critical Point, c, is not a point of inflection, it is a relative max or min:

If f"(c) > 0, the point is a min.

If f"(c) < 0, the point is a max.

The max may also be the end of the function, so check there to get every max/min.

Hint 53

Point C test

Front 54

Finding Absolute Maxs and Mins

Back 54

An absolute max or min is the highest or lowest point in a function, respectively.

You can find this by finding all the relative maxs and mins and choosing the ones with the highest and lowest y-values, respectively.

Front 55

Derivative Product Rule

Back 55

d(uv) = udv + vdu

Front 56

Tangent lines

Back 56

A tangent line is a linear line with the slope of the point it intersects on a function.

if a tangent line crosses the point n on a function f(x), the equation can be found easily using point-slope form:

(y-f(n))=f'(n)(x-n) <

Hint 56

What is a tangent line? What is the equation of a tangent line on a function, f(x), crossing through the point n?

Front 57

Normal lines

Back 57

A normal line is a linear line perpendicular to the line it intersects on a function; this means its slope is a negative reciprocal of the intersecting points' derivative.

if a normal line crosses the point n on a function f(x), the equation can be found easily using point-slope form:

(y-f(n))=-(1/f'(n))(x-n) <

Hint 57

What is a normal line? What is the equation of a normal line on a function, f(x), crossing through the point n?

Front 58

L'hopital

Back 58

Hint 58

In indeterminant cases only

Front 59

Derivative Quotient Rule

Back 59

Hint 59

remember, "voodoo"

Front 60

Chain Rule

Back 60

Hint 60

The Derivative of a function in a function

Front 61

Direct Substitution Law (limits)

Back 61

Hint 61

Front 62

Derivative of e^x

Back 62

d/dx(e^x) = e^x

Hint 62

simple

Front 63

What is a log?

Back 63

Hint 63

Opposite of exponants

Front 64

Adding logs of the same base

Back 64

Hint 64

adding >> multiplying

Front 65

Subtracting logs of the same base

Back 65

Hint 65

Subtracting >> dividing

Front 66

Multiplying a log (by an integer)

Back 66

Hint 66

multiplying >> exponant

Front 67

How do you change the base of a log to a new number?

Back 67

Hint 67

Involves division...

Front 68

How do you isolate an exponent of an expression?

Back 68

Hint 68

How can you turn B^n into n?

Front 69

What bases can a log have?

Back 69

if log(base x) is real, x MUST be greater than 0. The domain is (0,∞); x>0

Hint 69

Front 70

0/∞ = ?

Back 70

0/∞ = 0

Front 71

∞/0 = ?

Back 71

∞/0 = undefined

Front 72

∞/∞ = ?

Back 72

∞/∞ = indeterminant

Front 73

0/0 = ?

Back 73

0/0 = indeterminant

Front 74

∞ + ∞ = ?

Back 74

∞ + ∞ = ∞

Front 75

-∞ + ∞ = ?

Back 75

-∞ + ∞ = indeterminant

Front 76

∞*∞ = ?

Back 76

∞*∞ = ∞

Front 77

-∞-∞ = ?

Back 77

-∞-∞ = -∞

Front 78

0^0 = ?

Back 78

0^0 = indeterminant

Front 79

0^∞ = ?

Back 79

0^∞ = 0

Front 80

1^∞ = ?

Back 80

1^∞ = indeterminant

Front 81

0^-∞ = ?

Back 81

0^-∞ = undefined

Front 82

∞^0

Back 82

∞^0 = indeterminant

Front 83

what is the derivitive of dy/dx with respect to x?

Back 83

Hint 83

(d/dx)(dy/dx)

Front 84

what is point-slope form?

Back 84

Hint 84

uses a point and a slope

Front 85

What is the equation for the area under a curve?

Back 85

Hint 85

Front 86

What is the formula to estimate the area under a curve WITHOUT integrals?

Back 86

in the interval [a,b], the area under the curve will be calculated using areas of trapezoids, the "Y"'s are the placements of the trapezoids; the intervals. Decide the number of trapezoids, the more the better, find the trapezoids, then plug all the coords. and things into the equation.

Hint 86

use trapezoids