ab calculus question strategies
DESCRIPTION
A guide to approaching AB Calculus AP QuestionsTRANSCRIPT
When you see…
Find the zeros
You think…
To find the zeros...To find the zeros...
Find roots. Set function = 0 Factor or use quadratic equation if quadratic. Graph to find zeros on calculator.
You think…
When you see…
Show that f(x) is even
Even functionEven function
f (-x) = f ( x)
y-axis symmetry
You think…
When you see…
Show that f(x) is odd
Odd functionOdd function
. f (-x) = -f (x)
OR -f(-x) = f(x)
origin symmetry
You think…
When you see…
Show that exists lim ( )x a
f x
Show existsShow exists lim ( )x a
f x
Show that
xfxfaxax
limlim
You think…
When you see…
Find xfax
lim , calculator allowed
Find Limit with calculator
.
Use TABLE [ASK], find y-values for x-values
close to a from left and right
You think…
When you see…
Find xfax
lim , no calculator
Find limits, no calculatorSubstitute x = a
1) limit is value if b
c , incl. 0
0; 0cc
2) DNE for 0
b
3) 0
0 DO MORE WORK!
a) rationalize radicals b) simplify complex fractions c) factor/reduce d) known trig limits
1. 0
sinlim 1x
x
x
2. 0
1 coslim 0x
x
x
e) piece-wise fcn: check if RH = LH at break .
You think…
When you see…
Find limx
f x ,
calculator allowed
Find limit with calculator
.
Use TABLE [ASK], find y-values for large values
of x, i.e. 999999999999
You think…
When you see…
Find limx
f x ,
no calculator
Find limit, no calculator
Ratios of rates of changes
1)fast
DNEslow
2) 0slow
fast
3) same
ratiosame
of coefficients .
You think…
When you see…
Find horizontal asymptotes of f(x)
Find horizontal asymptotes of f(x)Find horizontal asymptotes of f(x)
Find
xfx lim
and
xf
x lim
You think…
When you see…
Find vertical
asymptotes of f(x)
Find vertical asymptotes of f(x)Find vertical asymptotes of f(x)
Find where limx a
f x
1) Factor/reduce xf and set denominator = 0
2) ln x has VA at x = 0
You think…
When you see…
Find the domain
of f(x)
Find the domain of f(x)Find the domain of f(x)
Assume domain is
, . Domain restrictions: non-zero denominators, Square root of non-negative numbers, Log or ln of only positive numbers Real-world constraints
You think…
When you see…
Show that f(x) is continuous
..f(x) is continuousf(x) is continuous
Show that
1) xfax
lim exists
2) af exists
3) afxfax
lim
You think…
When you see…
Find the slope of the tangent line to f(x) at x = a
Slope of tangent line
. Find derivative maf
When you see…
Find equation of the line tangent to f(x) at (a, b)
You think…
Equation of the tangent lineEquation of the tangent line
maf and use y b m x a
sometimes need to find b f a
You think…
When you see…
Find equation of the line normal to f(x) at (a, b)
Equation of the normal lineEquation of the normal line
Same as previous but
afm
1
You think…
When you see…
Find the average rate of change of f(x) at [a, b]
Average rate of change of f(x)Average rate of change of f(x)
Find
f (b) - f ( a)
b - a
You think…
When you see…
Show that there exists a c in ba, such that f c n
Intermediate Value Theorem (IVT)
Confirm that xf is continuous on ba, , then show that
( ) ( )f a n f b .
You think…
When you see…
Find the interval where f(x) is increasing
ff(x) increasing(x) increasing
Find xf , set both numerator and denominator to zero to find critical points, make sign chart of
xf and determine where xf is positive. Closed interval, unless f(x) DNE.
You think…
When you see…
Find the interval where the slope of f (x) is increasing
Slope of Slope of f f (x) is increasing(x) is increasing
Find the derivative of f ’(x) = f “ (x) Set numerator and denominator = 0 to find critical points Make sign chart of f “ (x)
Determine where f “ (x) is positive
Also, think f(x) is CONCAVE UP
You think…
When you see…
Find the instantaneous rate of change of f(x)
on [a, b]
Instantaneous rate of change of f(x)Instantaneous rate of change of f(x)
Find f ‘ ( a)
You think…
When you see…
Given s(t) (position function), find v(t)
Given position s(t), find v(t)Given position s(t), find v(t)
Find tstv
You think…
When you see…
Find f ’(x) by the limit definition
Find f Find f ‘‘( x) by definition( x) by definitionfrequently asked backwardsfrequently asked backwards
ax
afxfaf
h
xfhxfxf
ax
h
lim
or lim0
You think…
When you see…
Find the average velocity of a particle
on [a, b]
Find the average rate of change on Find the average rate of change on [a,b][a,b]
Find
v ta
b
dt
b as b s a b a
Depending on if you know s(t) or v(t)
You think…
When you see…
Given v(t), determine if a particle is speeding up at
t = k
Given v(t), determine if the particle is Given v(t), determine if the particle is speeding up at t = kspeeding up at t = k
Find v (k) and a (k). If signs match, the particle is speeding up. If signs opposite, the particle is slowing down
You think…
When you see…
Given a graph of
find where f(x) is
increasing
'( )f x
Given a graph of f ‘(x) , find where f(x) is Given a graph of f ‘(x) , find where f(x) is increasingincreasing
Make a sign chart of xf
Determine where xf is positive (above x-axis)
You think…
When you see…
Given a table of x and xf on selected values between a and b, estimate cf
where c is between a and b.
Straddle c, using a value, k, greater than c and a value, h, less than c.
so hk
hfkfcf
You think…
When you see…Given a graph of xf , find where xf has a relative maximum
Relative maximum
. Identify where 0f x crosses the x-axis from above to below OR where xf is discontinuous and jumps from above to below the
x-axis
You think…
When you see…Given a graph of xf ,
find where xf is concave down
Concave Down
. Identify where xf is decreasing
You think…
When you see…Given a graph of xf ,
find where xf has point(s) of inflection
Points of Inflection
. Identify where xf changes from increasing to decreasing or vice
versa
You think…
When you see…
Show that a piecewise function is differentiable at the point a where the function rule splits
Show a piecewise function is Show a piecewise function is
differentiable at x=adifferentiable at x=a
First, be sure that the function is continuous at
ax by evaluating a in both parts
Take the derivative of each piece and show that
xfxfaxax
limlim
You think…
When you see…
Given a graph of xf and 1h x f x , find 'h a
Derivative of Inverse Function
.
Find the point where a is the y-value on xf , sketch a tangent
line and estimate 'f b at the point,
then
1'
'h a
f b
You think…
When you see…
Given the equation for xf and 1h x f x , find
'h a
Derivative of the inverse of f(x)Derivative of the inverse of f(x)
Understand that the point ,a b is on h x so the point ,b a is on f x . So find b where f b a
1
''
h af b
You think…
When you see…Given the equation
for xf , find its derivative algebraically
Derivative Rules. 1) know product/quotient/chain rules 2) know derivatives of basic functions
a. Power Rule: polynomials, radicals, rationals
b. ;x xe b
c. ln ; logbx x
d. sin ;cos ; tanx x x
e. 1arcsin ;arccos ;arctan ;sin ;x x x x etc
You think…
When you see…Given a relation of x and
y, find dy
dx algebraically
Implicit Differentiation
Find the derivative of each term, using product/quotient/chain appropriately,
especially, chain rule: every derivative of y
is multiplied by dy
dx ; then group all dy
dx terms
on one side; factor out dy
dx and solve.
You think…
When you see…
Find the derivative off(g(x))
Chain RuleChain Rule
xgxgf
You think…
When you see…
Find the minimum value of a function
Minimum value of a functionMinimum value of a function
Solve 0f x or DNE, make a sign chart, find sign change from negative to positive for relative minimums and evaluate those candidates along with endpoints back into
xf and choose the smallest. NOTE: be careful to confirm that xf exists for any x-values that make 'f x
DNE.
You think…
When you see…
Find the minimum slope of a function
Minimum slope of a functionMinimum slope of a function
Solve " 0f x or DNE, make a sign chart, find sign change from negative to positive for
relative minimums and evaluate those candidates along with endpoints back into
'f x and choose the smallest. NOTE: be careful to confirm that xf exists
for any x-values that make "f x DNE.
You think…
When you see…
Find critical numbers
Find critical numbersFind critical numbers
Express xf as a fraction and solve for
numerator and denominator each equal to zero.
You think…
When you see…
Find the absolute maximum of f(x)
Find the absolute minimum of f(x)Find the absolute minimum of f(x)
Solve 0f x or DNE, make a sign chart, find sign change from positive to
negative for relative maximums and evaluate those candidates into xf , also
find xfx lim and xf
x lim ; choose the
largest.
You think…
When you see…
Show that there exists a c in ba, such that ' 0f c
Confirm that f is continuous and differentiable on the interval. Find k and j in ba, such that f k f j , then
there is some c in ,k j such that
f c 0.
Rolle’s Theorem
You think…
When you see…
Show that there exists a c in ba, such that 'f c m
Mean Value Theorem
Confirm that f is continuous and differentiable on the interval. Find k
and j in ba, such that
.f k f j
mk j
,
then there is some c in ,k j such that .f c m
You think…
When you see…
Find the range
of f(x) on [a, b]
Find the range of f(x) on [a,b]Find the range of f(x) on [a,b]
Use max/min techniques to find values at relative max/mins. Also compare f a and f b
(endpoints)
You think…
When you see…
Find the range
of f(x) on ( , )
Use max/min techniques to find relativemax/mins.
Then examine
limx
f x .
Find the range of f(x) onFind the range of f(x) on ,
You think…
When you see…Find the locations of relative extrema of
xf given both 'f x and "f x .
Particularly useful for relations of x and y where a sign chart would be difficult.
Second Derivative Test
.
Find where ' 0 OR DNEf x then check the
value of "f x there. If "f x is positive, xf
has a relative minimum. If "f x is negative, xf has a relative minimum.
You think…
When you see…
Find inflection points of xf algebraically.
Find inflection pointsFind inflection points
Express xf as a fraction and set both numerator and denominator equal to zero. Make sign chart of xf to find where
xf changes sign. (+ to – or – to +) NOTE: be careful to confirm that xf exists for any x-values that make "f x DNE.
You think…
When you see…
Show that the line bmxy is tangent to
xf at 1 1,x y
.y = mx+b is tangent to f(x) at a pointy = mx+b is tangent to f(x) at a point
Two relationships are required: same slope and point of intersection. Check that
1'm f x and that 1 1,x y is on both xf and the tangent line.
You think…
When you see…Find any horizontal
tangent line(s) to xf or a relation of x and y.
Horizontal tangent lineHorizontal tangent line
Write dy
dx as a fraction. Set the numerator equal to zero. NOTE: be careful to confirm that any values are on the curve. Equation of tangent line is y = b. May have to find b.
You think…
When you see…Find any vertical tangent
line(s) to xf or a relation of x and y.
Vertical tangent line to f(x)Vertical tangent line to f(x)
Write
dy
dx as a fraction. Set the denominator equal to zero.
NOTE: be careful to confirm that any values are on the curve.
Equation of tangent line is x = a. May have to find a.
You think…
When you see…
Approximate the value f(0.1) of by using the
tangent line to f at x = 0
Approximate f(0.1) using tangent line Approximate f(0.1) using tangent line to f(x) at x = 0to f(x) at x = 0
Find the equation of the tangent line to f using 11 xxmyy where 0fm and the point is 0,0 f . Then plug in 0.1 into this
line; be sure to use an approximate sign.
Alternative linearization formula: 'y f a x a f a
You think…
When you see…
Find rates of change for volume problems
Rates of Change of Volumes
.
Write the volume formula. Find dV
dt . Careful about product/ chain rules. Watch
positive (increasing measure)/negative (decreasing measure) signs for rates.
You think…
When you see…Find rates of change for Pythagorean Theorem
problems
Pythagorean Rates of Change
.
2 2 2x y z
2 2 2dx dy dz
x y zdt dt dt
; can reduce 2’s
Watch positive (increasing distance)/negative (decreasing distance) signs for rates.
You think…
When you see…
Find the average valueof xf on ba,
Average value of the functionAverage value of the function
Find 1
b
a
f x dxb a
You think…
When you see…
Find the average rate of change of xf on ba,
Average Rate of Change
.
f b f a
b a
You think…
When you see…
Given v(t), find the total distance a particle travels on [a, b]
Given v(t), find the total distance a Given v(t), find the total distance a particle travels on [a,b]particle travels on [a,b]
Find b
a
dttv
You think…
When you see…
Given v(t), find the change in position of a particle on [a, b]
Given v(t), find the change in position Given v(t), find the change in position of a particle on [a,b]of a particle on [a,b]
Find b
a
v t dt
You think…
When you see…
Given tv and initial position of a particle, find the position at t = a.
Given Given v(t)v(t) and the initial position of a and the initial position of a particle, find the position at particle, find the position at tt = = aa..
Find
0
0a
v t dt s
Read carefully: “starts at rest at the origin” means 0 0s and 0 0v
You think…
When you see…
dttfdx
d x
a
Fundamental TheoremFundamental Theorem
xf
You think…
When you see…
( )
g x
a
df t dt
dx
Fundamental Theorem, againFundamental Theorem, againChain Rule, tooChain Rule, too
( ) '( )f g x g x
You think…
When you see…
Find area using left Riemann sums
Area using left Riemann sumsArea using left Riemann sums
1210 ... nxxxxbaseANote: sketch a number line to visualize
You think…
When you see…
Find area using right Riemann sums
Area using right Riemann sumsArea using right Riemann sums
1 2 3 ... nA base x x x x
Note: sketch a number line to visualize
You think…
When you see…
Find area using midpoint rectangles
Area using midpoint rectanglesArea using midpoint rectangles
Typically done with a table of values. Be sure to use only values that are given. If you are given 6 sets of points, you can only do 3 midpoint rectangles. Note: sketch a number line to visualize
You think…
When you see…
Find area using trapezoids
Area using trapezoidsArea using trapezoids
nn xxxxxbase
A 1210 2...222
This formula only works when the base (width) is the same. Also trapezoid area is the average of LH and RH. If different widths, you have to do individual
trapezoids, 1 2
1
2A h b b
You think…
When you see…Describe how you can tell if rectangle or trapezoid approximations over- or under-
estimate area.
Over- or Under-estimates
Overestimate area: LH for decreasing; RH for increasing; and trapezoids for concave up
Underestimate area: LH for increasing; RH for decreasing and trapezoids for concave down
DRAW A PICTURE with 2 shapes.
.
You think…
When you see…
Given , find dxxfb
a
dxkxfb
a
Given area under a curve and vertical Given area under a curve and vertical shift, find the new area under the curveshift, find the new area under the curve
dxkdxxfdxkxfb
a
b
a
b
a
You think…
When you see…
Given , draw a
slope field dx
dy
Draw a slope field of dy/dxDraw a slope field of dy/dx
Use the given points
Plug them into dx
dy,
drawing little lines with theindicated slopes at the points.
You think…
When you see…
y is increasing proportionally to y
.y is increasing proportionally to yy is increasing proportionally to y
kydt
dy
integrating to
kty Ae
You think…
When you see…
Solve the differential equation …
Solve the differential equation...Solve the differential equation...
Separate the variables – x on one side, y on the other. The dx and dy must all be upstairs. Integrate each side, add C. Find C before solving for y,[unless ln y , then solve for y first and find A]. When solving for y, choose + or – (not both), solution will be a continuous function passing through the initial value.
You think…
When you see…
Given a base, cross sections perpendicular to
the x-axis that are squares
Semi-circular cross sections Semi-circular cross sections perpendicular to the x-axisperpendicular to the x-axis
The distance between the curves is the base of your square.
So the volume is 2b
a
f x g x dx
You think…
When you see…
Given the value of F(a) and the fact that the
anti-derivative of f is F, find F(b)
Given Given FF((aa)) and the that the and the that the anti-derivative of anti-derivative of ff is is FF, find , find FF((bb))
Usually, this problem contains an antiderivative you cannot do. Utilize the fact that if xF is the antiderivative of f,
then b
a
f x dx F b F a .
Solve for bF using the calculator to find the definite integral, then
( )b
a
F b f x dx F a
You think…
When you see…
Meaning of
b
a
f t dt
Meaning of the integral of f(t) from a to xMeaning of the integral of f(t) from a to x
The accumulation function –
Net (positive f(x) => total) amount of y –units for function xf
from x = a to x = b.
You think…
When you see…
Given v(t) and s(0), find the greatest distance from the origin of a particle on [a, b]
Given Given vv((tt)) and and ss(0)(0), find the greatest distance from , find the greatest distance from the origin of a particle on [the origin of a particle on [aa, , bb]]
Solve 0 OR DNEv t . Then integrate tv adding 0s to find ts . Finally, compare s(each candidate) and s(each endpoint). Choose greatest distance (it might be negative!)
When you see…
Given a water tank with g gallons initially being filled at
the rate of tF gallons/min and
emptied at the rate of tE gallons/min on 0,b , find
You think…
a) the amount of water in
the tank at m minutes
Amount of water in the tank at t minutesAmount of water in the tank at t minutes
0
m
g F t E t dt
You think…
b) the rate the water
amount is changing
at m
Rate the amount of water is changing at t = m
0
mdF t E t dt F m E m
dt
You think…
c) the time when the
water is at a minimum
The time when the water is at a minimumThe time when the water is at a minimum
Solve 0F t E t to find candidates, evaluate
candidates and endpoints as x a in
0
a
g F t E t dt , choose the minimum value
You think…
When you see…
Find the area between xf and g x with xf > g x on ba,
Area between f(x) and g(x) on [a,b]Area between f(x) and g(x) on [a,b]
dxxgxfAb
a
You think…
When you see…
Find the volume of the area between xf and g x with f x g x , rotated about the
x-axis.
Volume generated by rotating area between Volume generated by rotating area between f(x) and g(x) about the x-axisf(x) and g(x) about the x-axis
2 2b
a
A f x g x dx
You think…
When you see…
Given v(t) and s(0),
find s(t)
Given v(t) and s(0), find s(t)Given v(t) and s(0), find s(t)
0
0t
s t v x dx s
You think…
When you see…
Find the line x = c that divides the area under
f(x) on [a, b] into two equal areas
Find the x=c so the area under f(x) is Find the x=c so the area under f(x) is
divided equallydivided equally
dxxfdxxfb
c
c
a
1
2
c b
a a
f x dx f x dx
OR
You think…
When you see…
Find the volume given a base bounded by xf and g x with xf > g x and cross
sections perpendicular to the x-axis are semi-circles
Semi-circular cross sections Semi-circular cross sections perpendicular to the x-axisperpendicular to the x-axis
The distance between the curves is the diameter of your circle. So the volume
is 2
1
2 2
b
a
f x g xdx