Ap Calculus Bc

Front 1

A function is continuous if...

Back 1

1. f(a) exists
2. lim as x-> a of f(x) exists
3. lim as x->a of f(x)=f(a)

Front 2

Which functions are continuous everywhere?

Back 2

Polynomial, Rational, functions made up of continuous functions
i.e. f + g , f - g , f * g, f/g ( if g dne 0)

Front 3

Intermediate Value Theorem

Back 3

If a function f is continuous on the closed interval [a,b] and k is a number with f(a) =< k =< f(b), then there exists a number c in [a,b] such that f(c)=k

Front 4

Definition of a derivative (2)

Back 4

f'(x)= lim h->0 of f(x + h) - f(x) / h

Hint 4

f'(a) = lim x->a of f(x) - f(a) / (x-a)

Front 5

Sum & Difference Rules for derivatives

Back 5

You can add or subtract derivatives
(g +/- f)' = g' +/- f'

Front 6

Product Rule

Back 6

(fg)' = fg' + f'g

Front 7

Quotient Rule

Back 7

(f/g)' = gf' - fg' / g^2

g cannot = 0

Front 8

Derivatives of Trig functions

Back 8

(sinx)' = cosx
(cosx)' = -sinx
(tanx)' = sec^2x
(cotx)' = -csc^2x
(secx)' = secxtanx
(cscx)' = -cscxcotx

Front 9

Derivatives of Inverse Trig functions

Back 9

(arcsinx)' = 1/sqrt(1-x^2)
(arccosx)' = -1/sqrt (1-x^2)
(arctanx)' = 1/sqrt (1+ x^2)
(arccotx)' = -1/sqrt (1-x^2)

Front 10

Derivatives of Exponential and Logarithmic Functions

Back 10

Let u be a differentiable function.

(e^u)' = e^u
(a^u)' = a^u lnu (du/dx)

Front 11

Implicit Differentiation

Back 11

1. Separate y and x terms
2. Differentiate each term of the equation with respect to x
3. Factor out (dy/dx) on the left side of the equation
4. Solve for (dy/dx)

Front 12

Approximating a Derivative (2)

Back 12

If given a graph and asked to approximate a derivative...
For example, at x=3.
1. Use slope of line segment b/w x=3 and x=4
2. Use slope of line segment b/w x=2 and x=3
3. Use slope of a line segment b/w x=2 and x=4

Note: Method 3 will give you the avg of Methods 1 and 2

Hint 12

If given a table of values, you can use the definition of a derivative to calculate it

Front 13

Rolle's Theorem

Back 13

If:
1. f is continuous on a closed interval
2. f is differentiable on the open interval (a,b)
3. f(a) = f(b) = 0

then there exists a number c in (a,b) such that f'(c)=0

Front 14

Mean Value Theorem

Back 14

1. f is continuous on a closed interval [a,b]
2. f is differentiable on the open interval (a,b)

then there exists a number c in (a,b) such that f'(c)= (f(b)-f(a))/ (b-a)

Front 15

Extreme Value Theorem

Back 15

If f is a continuous function on a closed interval [a,b], then f has both a maximum and minimum value on the interval

Front 16

Test for Increasing/Decreasing Functions

Back 16

If f'(x) < 0 on [a,b] , then f(x) is decreasing on [a,b]
If f'(x) > 0 on [a,b] , then f(x) is increasing on [a,b]
If f'(x) = 0 on [a,b] , then f(x) is a constant on [a,b]

Front 17

First Derivative Test for Relative Extrema

Back 17

If f'(x) changes from pos to neg at x=c, then f has a relative maximum at c
If f'(x) changes from neg to pos at x=c, then f has a relative minimum at c

Front 18

Second Derivative Test for Relative Extrema

Back 18

If f'(c)=0 and f''(c)<0, then f(c) is a relative maximum
If f'(c)=0 and f''(c)> 0, then f(c) is a relative minimum
If f'(c)=0 and f''(c)=0, then f(c) is a relative maximum

Front 19

A point P is a point of inflection if...

Back 19

1. The curve has a tan line at P, and
2. The curve changes concavity at P (from concave upward to downward or from concave downward to upward)

(sign line of second derivative)

Front 20

General Procedure for Sketching Graph of a Function....

Back 20

1. Determine the domain (and range if possible)
2. Determine if the function has any symmetry (even, odd, periodic...)
3. Find f'(x) and f''(x)
4. Find all critical numbers (f'(x)=0 or undefined) and possible points of inflection (f''(x)=0 or undefined)
5. Using numbers from step 4, determine intervals on which to analyze f(x)
6. Setup a table using the intervals to
a. determine where f is increasing/decreasing
b. find relative and abs. extreme
c. find pts of inflection
d. determine concavity of f on each interval
7. Find any horizontal, vertical, or slant asymptotes
8. If necessary, find x-intercepts and y intercepts and a few selected points