CALC
A BRIEF GUIDE TO AP
AB
FOR THOSE WHO NEED A GOOD OVERVIEW BEFORE THE EXAM
Vital Approach
by CHRIS REDMOND WHERE IT COUNTS
TM
Limits &
Continuity 2
What is a Limit? A limit is the height a function intends to reach at a given x value, whether or not it actually reaches it We write:
lim f(x)=L x c Examples
A)
f(x)= x2-3x+2
lim f(x) = 6 x 4
This is the substitution method. Simply plug the number you’re approaching (4) in for the variable (x).
B)
x2-9 f(x)= x+3
=
lim f(x) = 7
(x+3)(x-3) x+3
x
10
= (x-3) This is the factoring method. Both the top and bottom of the fraction contain (x+3), so you can cancel those terms out and then use the substitution method.
When
a
Limit Does Not Exist
1.) f(x) approaches a different number from the right side of c and the left side at c 8
x lim x x
2.) f(x) increases/decreases without bound as x
1 lim x x 0
c
3
3.) f(x) oscillates between values as x
1 lim sin x 0 x
Properties
of
c
Limits
Let b and c be real numbers. Let n be a positive integer. Let f and g be functions with the following limits:
lim g(x)=K x c
lim f(x)=L x c Scalar Multiple
Quotient
lim bf(x) = bL x
f(x)
L K
lim g(x) = x c
c
Sum/Difference
Power
n n lim [f(x)] =L x c
lim [f(x) +_ g(x)]=L +_ K x c Product
lim [f(x)g(x)]=LK x c Theorem lim b=b x c
4
lim x=c x c
lim xn=cn
x
c
Theorem If p is a polynomial function and c is a real number,
lim p(x)=p(c) x c p(x) If r is a rational function given by r(x)= g(x) and c is a real number such that g(c)=0 then,
lim r(x)= x
c
p(c) g(c)
Theorem If f and g are functions such that xlimc g(x)=L and
lim f(x)=f(L) x
L
then
lim f(g(x))=f(L) x
c
Theorem
lim sin x = sin c
lim cot x = cot c x c
lim cos x = cos c x c
lim sec x = sec c x c
lim tan x = tan c x c
lim csc x = csc c x c
x
c
Theorem Let c be a real number and let f(x)=g(x) for all x=c in an open f(x) exists, the lim g(x) also exists and interval containing c. If lim x c x c lim f(x) = lim g(x) x c x c
5
Squeeze Theorem If h(x) f(x) g(x) for all x in an open interval containing c, h(x) =L= lim g(x), then lim f(x)=L except possibly at c itself, and if lim x c x c x
c
Theorem
lim x 0
sin x x
1-cos x x
lim x 0
=1
=0
cos x-1 x
lim x 0
=0
What is Continuity? A function is continuous on an open interval (a,b) if for every x in the interval, there is a y.
Continuity At
a
Point
A function is defined at c if: 1) f(x) is defined f(x) exists 2) lim x c f(x) = f(c) 3) lim x c
Discontinuities Removeable
can be made continuous by defining or redefining f(c)
6
Nonremoveable
cannot be made continuous by defining or redefining f(c)
Intermediate Value Theorem If f is continuous on [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k
Limits
of Infinity
If r is a positive rational number and c is any real number then c xr x
=0
8
lim
If xr is defined for x<0
lim -
8
x
c xr
=0
Practice Problems Find the following limits.
1)
lim x
2
x2-4 x2 +4
4-x2 x2-1
3)
lim
x-3 x -2x-3
x
x
8
2)
lim
3
2
7
Derivatives
8
What is a Derivative? A derivative is the slope of a function. At any x in the domain of the function y=f(x), the derivative is defined as f(x+ x)-f(x) y OR x x x 0 x 0
lim
lim
The function is said to be differentiable at every x for which this limit exists, and its derivative may be denoted by f’(x), y’, dy , or Dxy. Frequently x is replaced by h or some other symbol. dx The derivative of y=f(x) at x=a, denoted by f’(a) or y’(a), may be defined as follows: f(a+h)-f(a) h f’(a) = h 0
lim
Basic Derivative Formulas da =0 dx d au=adu dx dx d n n-1 dx x =nx (Power Rule) d _ d _dv dx (u+v)=dx u+ dx d dv du dx (uv)=u dx +v dx (Product Rule) d u v du-u dv dx v = dx 2 dx v=0 (Quotient Rule) v d sin u= cos udu dx dx
( )
9
d cos u= -sin u du dx dx d tan u=sec2u du dx dx d cot u= -csc2u du dx dx d sec u= sec u tan u du dx dx d du csc u= -csc u cot u dx dx d 1 du ln u= u dx dx d eu=eu du dx dx d au=au ln a du dx dx
Constant Multiple Rule d [cf(x)]=c d [f(x)] dx dx
Chain Rule d [f(g(x))=f”(g(x))g’(x) dx
General Power Rule d [u(x)]n=n[u(x)]n-1 du dx dx
10
Implicit Diferentiation Implicit Differentiation comes into play when a functional relationship between x and y is defined by an equation of the form F(x, y)=0. For example: x2+y2-9=0 y2-4x=0 cos (xy)=y2-5 d [y3]=3y2 dy dx dx
Use chain rule (let y=f(x))
Examples
A)
d xy2 = x2ydy +y2 = 2xydy +y2 dx dx dx
B) ddx [x y ] = x 2ydydx +3x y = 2x ydydx+3x y 3 2
Derivative
3
2 2
3
of the Inverse of a
2 2
Function
The derivative of the inverse of a function at a point is the reciprocal of the derivative of the function at the corresponding point 1 (f-1)’(x)= 1-1 (f-1)’(b)= f’(a) OR f’(f (x))
Mean Value Theorem If the function f(x) is continuous at each point on the closed interval a x b and has a derivative at each point on the open interval a x b, then there is at least one number c, a c b, such that f(b)-f(a) =f’(c). This important theorem b-a relates average rate of change and instantaneous rate of change.
11
Critical Number Let f be defined at c. If f’(c)=0 or if f is not differentiable at c, then c is a critical number.
Extreme Value Theorem If f is continuous on a closed interval [a,b] then f has both a max and a min in that interval. These values can be the endpoints. continuous open interval no max no min
discontinuous closed interval has max no min
Rolle’s Theorem Let f be continuous on [a,b] and differentiable on (a,b). If f(a)=f(b) then there is at least one number c such that f’(c)=0.
12
First Derivative Test Assume that f(x) is differentiable and let c be a critical number of f(x) If f’(x) changes from + to - at c THEN f(c) is a real max If f’(x) changes from - to + at c THEN f(c) is a real min
Test
for
Concavity
Assume that f”(x) exists for all in (a,b) If f”(x) > 0 THEN the graph is concave up If f”(x) < 0 THEN the graph is concave down
Test
for Inflection
Point
If f”(c)=0 and f”(x) changes sign at x=c, then there is an inflection point at x=c
Practice Problems Find y’.
1)
y=x5tan x
2)
y=
3)
y = 3x3+4y2
2-x 3x+1
13
Antiderivatives
14
What is a Antiderivative? The antiderivative or indefinite integral of a function f(x) is a function F(x) whose derivative is f(x). Since the derivative of a constant equals zero, the antiderivative of f(x) is not unique; that is, if F(x) is an integral of f(x), then so is F(x) + C, where C is any constant. The arbitrary constant C is called the constant of integration. f(x) dx=F(x) + C integrand
constant of integration
variable of integration
Basic Antiderivative Formulas kf(x) dx=k f(x) dx
(k=0)
csc u cot u du = -csc u +C
_ [f(x)+g(x)] dx = f(x) dx _+ g(x) dx
sec u du = ln sec u + tan u +C
n+1 un du = un+1 +C (n = -1)
csc u du = ln csc u + cot u +C
du =ln u +C u cos u du = sin u+C
eu du = eu +C au du =
au +C ln a
(a>0, a=1)
sin u du = -cos u+C tan u du = ln sec u +C cot u du = ln sin u +C sec2 u du = tan u +C csc2 u du = -cot u+C sec u tan u du = sec u +C
15
integration
Rules _ g(x)dx _ g(x)dx = f(x)dx + f(x) +
0dx = C kdx = kx+C
n+1 xndx = x +C= 1 xn+1+C n+1 n+1
kf(x)dx = k f(x)dx
Examples
A)
3
B)
1 -1 -1 -1 -2 x2 dx = x dx = 1 x +C = x +C
1 x dx= x1/3dx = 4/3 x4/3+C = 34 x4/3+C
U-Substitution 1. Look for a piece of the function whose derivative is also in the function. If youâ&#x20AC;&#x2122;re not sure what to use, try the denominator or something being rasied to a power in the function. 2. Set u equal to that piece of the funciton and take the derivative with respect to nothing. 3. Use your u and du expressions to replace parts of the original integral, and your new integral will be much easier to solve.
Examples
A) B) 16
5 2x(x2+1)4dx = u4du= u +C = 1 (x2+1)5+Ct 5 5 u = x2+1 du = 2xdx
2 3x2 x3+1 dx= u1/2du = 3 u3/2+C = 23 (x3+1)3/2+C u = x3+1 du = 3x2dx
Definite Integrals To find exact areas under curves, use definite integrals. The 3 area beneath x2+1 on the interval [0,3] is equal to 0 (x2+1) dx.
Properties a
b
a
b a b a
b
kf(x)dx = k
f(x)dx =
b
Definite Integrals
f(x)dx = 0
a
a
of
f(x)dx =
_
b a
a
f(x)dx
f(x)dx
c
f(x)dx + a
[f(x)+_ g(x)]dx =
b a
b c
f(x)dx
f(x)dx +_
b a
a<c<b g(x)dx
Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b] then b a
f(x)dx = F(b)-F(a) b
b
f(x)dx = F(x) a a
Practice Problems Answer these questions.
1)
(3x2-2x+3)dx
2)
(2-3x)5dx
3)
4-2t dt
17
Answers to
Problems 18
Limits & Continuity
1) 0 2) -1 3) 41
Derivatives
1)
By the Product Rule: y’=x5(tan x)’+(x5)’(tan x) y’=x5sec2 x+5x4tan x
2) By the Quotient Rule:
(3x+1)(-1)-(2-x)(3) (3x+1)2 7 y’= - (3x+1)2 y’=
3) 9x +8ydx 2
dy
Antiderivatives
1) x3-x2+3+C -1 2) 18 (2-3x) +C 3) -12 4-2t (-2dt) = -12 (4-2t) 3/2 6
3/2
+C
u=4-2t du=-2
19
What is a limit? Page 3 What is a derivative? Page 9 What is the first derivative test? Page 13 What is an antiderivative? Page 15 What are definite integrals? Page 17