AP Calc AB Formulas/Tips and Tricks

Find zeros of f(x)

Set f(x)=0 and solve for x

Find intersection of f(x) and g(x)

Set f(x)=g(x) and solve for x

f(x) is EVEN

f(-x)=f(x)

f(x) is ODD

f(-x)=-f(x)

Find domain of f(x)

* Set the denominator=0 (rational function)
* Set terms inside radicand (under square root) ≥ 0
* Set terms inside parenthesis (argument) > 0 (logarithm)

When do limits NOT exist?

1. *lim* f(x) \[as x→c-\] ≠ *lim* f(x) \[as x→c+\]
2. f(x) approaches ∞ \[as x→c\]

What is indeterminate form?

0/0, ∞/∞, 0⋅∞ (hole in graph)

***special trig limit***

*lim* (sin**x**)/**x** \[as ^^x→0^^\] = ???

1

***special trig limit***

*lim* (1-cos**x**)/**x** \[as ^^x→0^^\] = ???

0

When is a function **continuous**? (==definition of continuity==)

1. f(a) is defined and a real #
2. *lim* f(x) \[as x→a-\] = *lim* f(x) \[as x→a+\] … therefore, *lim* f(x) \[as x→a\] **exists**
3. f(a) = *lim* f(x) \[as x→a\]

==Intermediate Value Theorem (IVT)==

If our function f(x) is **continuous** on \[a,b\] (the interval), and K is in between f(a) and f(b), then there must be a c-value on (a,b) such that f(c)=K

How do you solve ***infinite limits***?

i.e. *lim* f(x) \[as x→c\] = #/0

Determine if f(x) tends towards +/- ∞

* Find VA
* Make a chart of x and f(x)
* Plug in x-values leading up to **vertical asymptote** and see if f(x) increases or decreases

How do you solve ***limits at infinity***?

i.e. *lim* f(x) \[as x→ +/- ∞\] = L (**horizontal asymptote**)

Find L, or the HA

* Try direct substitution FIRST
* Convert each term of the fraction to (1/x) or a variation of it
* Divide each term by the HIGHEST POWER that x is raised to in the *denominator*
* When you have a radical, divide by highest power OUTSIDE radical

==**Definition of a Derivative**== at ANY POINT

*lim* $f(x + Δx)-f(x) / Δx$ \[as Δx→0\] = f’(x) … slope of tangent line

==**Definition of a Derivative**== at x=c

*lim* $f(x)-f(c) / (x-c)$ \[as x→c\] = f’(c) … slope of tangent line at x=c

Equation of Tangent Line to f(x) at x=c

y - $**f**$**(c)** = **f’(c)**⋅(x - $**c**$)

If a function is __differentiable__, does that mean it’s __continuous__?

YES

If a function is __continuous__, does that mean it’s __differentiable__?

NO \~ think sharp turns/cusps

What makes a function **NOT** differentiable?

1. Discontinuous
2. Sharp turns/cusps
3. Vertical tangent line, or VA

Rule for **basic differentiation**

d/dx \[xⁿ\] = n⋅xⁿ⁻¹

d/dx \[eˣ\] = ???

eˣ

d/dx \[lnx\] = ???

1/x

d/dx \[sinx\] = ???

cosx

d/dx \[cosx\] = ???

\-sinx

d/dx \[tanx\] = ???

sec²x

d/dx \[cotx\] = ???

\-csc²x

d/dx \[secx\] = ???

secxtanx

d/dx \[cscx\] = ???

\-cscxcotx

“Find the x-values at which f(x) has a **horizontal** tangent line”

SLOPE = 0

f’(x) = 0 … set derivative to 0 and find x

***product rule***

d/dx \[f(x)g(x)\] = ???

f(x)g’(x) + g(x)f’(x)

1 d2 + 2 d1

***quotient rule***

d/dx \[f(x)/g(x)\] = ???

(g(x)f’(x) - f(x)g’(x)) / (g(x))²

(low dhigh - high dlow) / (low)²

***chain rule***

d/dx \[f(g(x))\] = ???

f’(g(x))⋅g’(x)

deriv of outer(leave inner)⋅deriv of inner

ln(1) = ???

0

ln(e) = ???

1

ln(ab) = ???

lna + lnb

ln(a/b) = ???

lna - lnb

lnaⁿ

n⋅lna

d/dx \[ln(u)\] = ???

u’/u

d/dx \[eᵘ\] = ???

eᵘ⋅u’

Position (==partical motion==)

s(t) = -16t² + Vot + So

Height above ground, water, etc (ft)

Velocity (==partical motion==)

v(t) = s’(t) = -32t + Vo

Rate of change of __position__ over __time__ (ft/sec)

Acceleration (==partical motion==)

a(t) = v’(t) = s”(t) = -32

Change in __velocity__ over __time__ (ft/sec²)

Average Velocity (==particle motion==)

s(b)-s(a) / (b-a)

change in position / change in time

Δx / Δt

Steps for ***implicit differentiation*** (w/ respect to x)

1. Always differentiate both sides with respect to x
2. Group all the dy/dx terms on 1 side, and everything else on the other side
3. Factor out dy/dx
4. Divide to solve for dy/dx

What is a ***related rate***?

Rates of change of 2 or more variables at the same time.

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Steps to solve ***related rate*** problems?

1. Make a sketch (label variables & any length/quantity that is CONSTANT)
2. Write out what we __KNOW__ (given) and what we __WANT__ (asked to find)
3. Write an equation that relates all variables
4. Differentiate implicitly w/ respect to TIME
5. Substitute known values & solve for WANT (required rate of change)
6. Write a conclusion that puts the answer back in terms of the problem & includes the *what*, *when*, *how much*


1. The ___ is changing by ___ when ___

Where are the only 2 places where ***absolute extrema*** can occur?

1. endpoints
2. critical points (f’(x)=0, und)

How do you find ***Absolute Max/Min***?

==THE CANDIDATES TEST==!!!

* find critical points
* find y-values at critical points AND end points
* determine abs max/min

==Mean Value Theorem (MVT)==

If a function is continuous AND differentiable on \[a,b\] (interval), then there must be some c-value on (a,b) such that f(b)-f(a) / (b-a) = f’(c).

\* IN OTHER WORDS: *slope of the secant* = *slope of the tangent*; *the average ROC* = *the instantaneous ROC*

How do you find the **slope of a secant**, or the **average ROC**?

f(b)-f(a) / (b-a) = answer

Relative Maximum

f’ is changing from + to -

Relative Minimum

f’ is changing from - to +

How do you find ***Relative Max/Min*** using the ==**1ST DERIVATIVES TEST**==?

1. Find critical points
2. Create a table w/ x-value intervals based on the critical points
3. Test values to see if f’ is + or -
4. Draw a conclusion w/ justification from the table

Table Connecting f, f’, f”

How do you find ***concavity intervals***?

\* Same process as ==1st Derivatives Test==!!! (only use f” instead, and determine if f is inc/dec)

Point of Inflection (POI)

Where the graph of f changes concavity; where f” changes signs (+/-)

How do you find ***Relative Max/Min*** using the ==**2ND DERIVATIVES TEST**==?

If f’(c)=0, and…

* f”(c)>0, then **REL MIN** at x=c (concave up)
* f”(c)<0, then **REL MAX** at x=c (concave down)

Rule for **basic integration**

∫xⁿ dx = (xⁿ⁺¹ / n+1) +c

∫cosx dx = ???

sinx +c

∫sinx dx = ???

\-cosx +c

∫secxtanx dx = ???

secx +c

∫cscxcotx dx = ???

\-cscx +c

∫sec²x dx = ???

tanx +c

∫csc²x dx = ???

\-cotx +c

“Solve the differentiable equation”

Find f(x) with c as a #

When finding area using **Riemann Sums**, how will we determine the height of each rectangle?

* Determined by the y-value of f(x)
* Depends on if we’re using *right*, *left*, *midpoint*

Formula for Riemann Sums using rectangles

A = b₁h₁ + b₂h₂ + …

Formula for Riemann Sums using trapezoids

A = ½(b₁+b₂)h₁ + ½(b₂+b₃)h₂ + …

Justification for **OVER/UNDER-ESTIMATE** w/ Riemann Sum rectangles

Because f(x) is inc/dec

Justification for **OVER/UNDER-ESTIMATE** w/ Riemann Sum trapezoids

Because f(x) is concave up/down

Trapezoidal Riemann *under-estimate*?

concave DOWN

Trapezoidal Riemann *over-estimate*?

concave UP

Integration units?

multiply

Derivative units?

divide

Average Value for Integrals

1/(b-a) ∫$f(x)$ dx \[from a→b\]

\
\* what goes IN the integrand is what you are taking the average of!

==Net Change Theorem==

F(b) = F(a) + ∫F’(x) dx \[from a→b\]

final amount = total change in amount from a to b

==Fundamental Theorem of Calculus (FTC)==

∫f’(x) dx \[from a→b\] = f(b) - f(a)

==2nd Fundamental Theorem of Calculus (2nd FTC)==

d/du ∫f(t) dx \[from a→u\] = f(u)⋅u’

\
\* taking the derivative of an integral

a=constant, u=function in terms of x

Tips for finding “u” in **U-Substitutions**

1. Functions being raised to a power
2. Arguments of trig functions that aren’t just “x” or “θ”
3. Trig function whose derivative is also in integrand

∫(1/x) dx = ???

ln|x| +c

∫(1/u) du = ???

ln|u| +c

Tips for success when **integrating natural log (ln)**

1. Use log properties to simplify
2. Most times, denom=u
3. Use __LONG DIVISION__ if degrees are equal OR num>denom
4. If num<denom, use regular u-sub

∫eᵘ du = ???

eᵘ +c

How do you find inverses?

Switch x & y, then solve for y

Rule for **derivative of an inverse**

g’(x) = ???

1 / (f’(g($x$))

$x$ = x-value of inverse, y-value of original

y = aᵘ

y’ = ???

(lna)⋅aᵘ⋅u’

y=logₐu

u’ / (lna)u

Steps for ==**Natural Log Differentiation**==

\* with $y = aᵘ$, use when both a and u are __FUNCTIONS__ (have x-variable)

1. Take natural log of both sides
2. Simplify w/ log properties
3. Take derivative of both sides w/ respect to x
4. Multiply both sides by “y”

∫aᵘ du = ???

aᵘ / (lna) +c

d/dx \[arcsinu\] = ???

u’ / √1-u²

d/dx \[arctanu\] = ???

u’ / 1+u²

d/dx \[arcsecu\] = ???

u’ / |u|⋅√u²-1

d/dx \[arccosu\] = ???

\-u’ / √1-u²

d/dx \[arccotu\] = ???

\-u’ / 1+u²

d/dx \[arccscu\] = ???

\-u’ / |u|⋅√u²-1

∫du / √a²-u² = ???

arcsin(u/a) +c

∫du / a²+u² = ???

(1/a)arctan(u/a) +c

∫du / u⋅√u²-a² = ???

(1/a)arcsec(|u|/a) +c

Tips to **Integrate Inverse Trig**

1. Multiply by equivalent form of 1
2. Separate into 2 fractions
3. If you have (constant/polynomial), COMPLETE THE SQUARE!
4. Add and subtract the same quantity

How to solve differential equations using **Separation of Variables**

\*must have product or quotient of x and y

1. Rewrite equation to get just dy & y on one side, and just dx & x on the other
2. Integrate both sides
3. Plug in the initial condition (a,b) to get your +c value
4. Then solve for y