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When you see…
A1. Find the zeros
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A1 To find the zeros...A1 To find the zeros...
Set function = 0
Factor or use quadratic equation if quadratic.
Graph to find zeros on calculator.
When you see…
A2. Find intersection of f(x) and g(x)
You think…
A2 To find the zeros...A2 To find the zeros...
Set f(x) = g(x) and solve (often on calculator).
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A3 Show that f(x) is even
A3 Even functionA3 Even function
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A4 Show that f(x) is odd
A4 Odd functionA4 Odd function
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A5 Find the domain
of f(x)
A5 Find the domain of f(x)A5 Find the domain of f(x)
• Assume domain is (-∞,∞).
• Restrictable domains:
–Denominators ≠ 0
–Square roots of only non negative #s
– log or ln of only positive #s
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A6 Find vertical
asymptotes of f(x)
A6 Find vertical asymptotes of f(x)A6 Find vertical asymptotes of f(x)
Express f(x) as fraction, with
numerator, denominator in factored
form. Reduce if possible.
Then set denominator = 0
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A7 If continuous function f(x) has f(a) < k and f(b) > k,
explain why there must be a value c such that a<c<b and
f(c) = k.
A7 Find f(c) = k where a<c<bA7 Find f(c) = k where a<c<b
This is the Intermediate Value Theorem.
We usually use it to find zeros between positive and negative function values, but it could be used to find any y-value between f(a) and f(b).
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B1 Find lim ( )x a
f x
B1 Find B1 Find lim ( )x a
f x
Step 1: Find f(a). If zero in denom, step 2
Step 2: Factor numerator, denominator and
reduce if possible. Go to step 1. If
still zero in denom, check 1-sided
limits. If both + or – infinity, that is
your answer. If not, limit does not
exist (DNE)
limx a
f x limx a
f x
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B2 Find where f(x) is a piecewise function.
lim ( )x a
f x
B2 Show existsB2 Show exists
(Piecewise)(Piecewise)
lim ( )x a
f x
Check 1-sided limits . . . .
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B3 When you see…
Show that f(x) is continuous
..B3B3 f(x) is continuousf(x) is continuous
limx a
f (x)
f (a)
limx a
f (x) f (a)
( limx a
f (x) limx a
f (x))1) exists
2) exists
3)
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B4 When you see…
Find )x(flimand)x(flimxx
B4 FindB4 Find
Express f(x) as a fraction, Express f(x) as a fraction, determine highest power. determine highest power.
If in denominator, limit = 0If in denominator, limit = 0If in numerator, lim = If in numerator, lim = ++
)x(flimand)x(flimxx
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B-5 Find horizontal asymptotes of f(x)
Find and
B5 Find horizontal asymptotes of f(x)B5 Find horizontal asymptotes of f(x)
limx
f (x)
limx
f (x)
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C1 Find f ’(x) by definition
C1Find f C1Find f ‘‘( x) by definition( x) by definition
f x limh0
f x h f x h
or
f x limx a
f x f a x a
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C2 Find the average rate of change of f(x) at [a, b]
C2 Average rate of change of f(x)C2 Average rate of change of f(x)
Find
f (b) - f ( a)
b - a
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C3 Find the instantaneous rate of change of f(x)
at a
C3 Instantaneous rate of change of f(x)C3 Instantaneous rate of change of f(x)
Find f ‘ ( a)
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C4 Given a chart of x and f(x) on selected values between a and b, estimate where c is between a and b.
f (c)
C4 Estimating f’(c) between a and b
Straddle c, using a value of k greater than c and a value h less than c.
So
f '(c) f (k) f (h)
k h
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C5 Find equation of the line tangent to f(x) at (x1,y1)
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C5 Equation of the tangent lineC5 Equation of the tangent line
Find slope m = f ’(x). Use point (x1 , y1)
Use Point Slope Equation:
y – y1 = x – x1
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C6 Find equation of the line normal to f(x) at (a, b)
C6 Equation of the normal lineC6 Equation of the normal line
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C7 Find x-values where the tangent line to f(x) is
horizontal
C7 Horizontal tangent lineC7 Horizontal tangent line
Write xf as a fraction.
Set the numerator equal to zero
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C8 Find x-values where the tangent line to f(x) is
vertical
C8 Vertical tangent line to f(x)C8 Vertical tangent line to f(x)
Write f ’(x) as a fraction. Set the denominator equal to zero.
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C9 Approximate the value of f (x1 + a) if you know
the function goes through (x1 , y1)
C9 C9 Approximate the value of (x1 + a)
Find the equation of the tangent line to f using y-y1 = m(x-x1). Now evaluate at x = x1+a. Note: The closer to a is to x1, the better the approximation.
Note: Can use f’’, concavity to tell if it is an under- or overestimate.
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C10 Find the derivative off(g(x))
C10 Find the derivative of f(g(x))C10 Find the derivative of f(g(x))
Composition of functions!
Chain Rule!
f’(g(x)) · g’(x)
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C11 The line y = mx + b is tangent to f(x) at (x1, y1)
C11 y = mx+b is tangent to f(x) at (a,b)y = mx+b is tangent to f(x) at (a,b)
Two relationships are true:
1) The function and the line have the same slope at x1: (m=f ’(x))
2) The function and line have same y-value at x1
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C12 Find the derivative of g(x), the inverse to f (x)
at x = a
C12 Derivative of g(x), the inverse of C12 Derivative of g(x), the inverse of f(x) at x=af(x) at x=a
On g use (a, Q)
On f use (Q, a) Find Q-value
So )Q(f
1)a(g
C12 Derivative of g(x), the inverse of C12 Derivative of g(x), the inverse of f(x) at x=af(x) at x=a
Interchange x with y.
Plug your x value into the inverse relation and solve for y
Solve for implicitly (in terms of y)
Finally plug that y into
dy
dx
dx
dy
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C13 Show that a piecewise function is
differentiable at the point a where the function rule
splits
C13 Show a piecewise function is C13 Show a piecewise function is
differentiable at x=adifferentiable at x=a
limx a
f '(x) limx a
f '(x)
Be sure the function is continuous at x = a
Take the derivative of each piece and show that
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D1 Find critical values
of f(x)
D1 Find critical valuesD1 Find critical values
Express f ´(x) as a fraction, simplify Set both numerator and denominator = 0 Use the x-values to analyze.
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D2 Find the interval(s) where f(x) is
increasing/dec.
f ’(x) < 0 means decreasing
D2 fD2 f(x) increasing(x) increasing
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D3 Find points of relative extrema of f(x)
D3 Find relative extremaD3 Find relative extrema
Sign chart for f ´(x) – must change from pos. to neg. for relative max, and neg. to pos. for relative min. (First Derivative Test) OR if f ′(c) = 0 and f ′′(c) is positive, f(x) is concave UP, so there is a min at x = c. If f ′′(c) is negative, then f(x) is concave DOWN, with a max at x = c. (Second Derivative Test)
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D4 Find inflection points
D4 Find inflection pointsD4 Find inflection points
Express f (x) as a fraction Set numerator and denominator = 0 Make a sign chart of f ″(x) Find where it changes sign ( + to - ) or ( - to + )
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D5 Find the absolute maximum of f(x) on [a, b]
(or minimum)
D5 Find the absolute max/min of f(x)D5 Find the absolute max/min of f(x)
1) Make a sign chart of f ’(x)
2) Find all relative maxima and plug into f(x) (or relative minima)
3) Find f(a) and f(b)
4) Choose the largest (or smallest)
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D6 Find the range
of f(x) on ( , )
D6 Find the range of f(x) onD6 Find the range of f(x) on
Use max/min techniques to find relative max/mins
Then examine
limx
f (x)
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D7 Find the range
of f(x) on [a, b]
D7 Find the range of f(x) on [a,b]D7 Find the range of f(x) on [a,b]
Use max/min techniques to find relative max/mins
Then examine f(a), f(b)
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D8 Show that Rolle’s Theorem holds on [a, b]
D8 Show that Rolle’s Theorem holds D8 Show that Rolle’s Theorem holds on [a,b]on [a,b]
Show that f is continuous and differentiable on the interval
If f(a)=f(b), then find some c in [a,b] such that f ’(c)=0
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D9 Show that the Mean Value Theorem holds
on [a, b]
D9 Show that the MVT holds on [a,b]D9 Show that the MVT holds on [a,b]
f '(c) f (b) f (a)
b a
Show that f is continuous and differentiable on the interval. Then find some c such that
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D10 Given a graph of
find where f(x) is
increasing/decreasing
'( )f x
D10 Given a graph of f ‘(x) , find where f(x) D10 Given a graph of f ‘(x) , find where f(x) is increasing/decreasingis increasing/decreasing
Make a sign chart of f’(x) and determine where f’(x) is
positive/negative
(increasing/decreasing)
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D11 Determine whether the linear approximation
for f(x1 +a) is over- or underestimate of actual
f(x1 + a)
D11 Determine whether f(xD11 Determine whether f(x11 + a) is + a) is
over- or underestimateover- or underestimate
See C9 above. Find f(x1 +a). Find f″ on an interval containing x1.
If concave up, underestimate. If concave down, overestimate.
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D12 Find the interval where the slope of f (x) is
increasing
D12 Slope of D12 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 ″ is positive
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D13 Find the minimum slope of a function
D13 Minimum slope of a functionD13 Minimum slope of a function
Make a sign chart of f ′(x) = f ″ (x)
Find all the relative minimums, where f ″
Changes from neg. to positive. Evaluate
f ′(x) here, and f ′(a) and f ′(b), choose smallest value.
(do reverse for maximum slope)
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E1 Find area using left Riemann sums
E1 Area using left Riemann sumsE1 Area using left Riemann sums
A=base[x0+x1+x2…+xn-1]
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E2 Find area using right Riemann sums
E2 Area using right Riemann sumsE2 Area using right Riemann sums
A=base[x1+x2+x3…+xn]
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E3 Find area using midpoint rectangles
E3 Area using midpoint rectanglesE3 Area 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.
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E4 Find area using trapezoids
E4 Area using trapezoidsE4 Area using trapezoids
This formula only works when the base is the same. If not, you must do individual trapezoids. I would EXPECT this!
]xx2...x2x2x[2
baseArea n1n210
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E8 Meaning of
dttfx
a
E8 Meaning of the integral of f(t) from a to xE8 Meaning of the integral of f(t) from a to x
The accumulation function
accumulated area under the function f(x) starting at some constant a and ending at x
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E9 Given ,find dxxfb
a
dxkxfb
a
E9 Given area under a curve and E9 Given area under a curve and vertical shift, find the new area under vertical shift, find the new area under
the curvethe curve
b
a
b
a
b
a
kdxdx)x(fdx]k)x(f[
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E10 Given the value of F(a) and the fact that
the
anti-derivative of f is F, find F(b)
E10 Given E10 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 take. Utilize the fact that if F(x) is the antiderivative of f, then
So, solve for F(b) using the calculator to find the definite integral.
F(x)dx F(b) F(a)a
b
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E11
d
dxf (t)dt
a
x
E11 Fundamental TheoremE11 Fundamental Theorem
2nd FTC: Answer is f(x)
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d
dxf (t)dt
a
u
E12
E12 Fundamental Theorem, againE12 Fundamental Theorem, again
2nd FTC: Answer is
f (u)du
dx
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F2 Find the area between curves f(x) and
g(x) on [a,b]
F2 Area between f(x) and g(x) on [a,b]F2 Area between f(x) and g(x) on [a,b]
Assuming that the f curve is above the g curve
A [ f (x) g(x)]dxa
b
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F3 Find the line x = c that divides the area
under f(x) on [a, b] into two equal areas
F3 Find the x=c so the area under f(x) F3 Find the x=c so the area under f(x)
is divided equallyis divided equally
c
a
b
c
dx)x(fdx)x(f
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F5 Find the volume if the area between the
curves f(x) and g(x) is rotated about the x-axis
F5 Volume generated by rotating area F5 Volume generated by rotating area between f(x) and g(x) about the x-axisbetween f(x) and g(x) about the x-axis
Assuming that the f curve is above the g curve
A [( f (x))2 (g(x))2]dxa
b
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F6 Given a base, cross sections perpendicular to
the x-axis that are squares
F6 Square cross sections F6 Square cross sections perpendicular to the x-axisperpendicular to the x-axis
The area between the curves is typically the base of the square so the volume is
(base2)dxa
b
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F7 Solve the differential equation …
F7 Solve the differential equation...F7 Solve the differential equation...
Separate the variables
x on one side, y on the other. The dx and dy must all be upstairs. Then integrate (+c), solve for y . . . .
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F8 Find the average value of f(x) on [a,b]
F8 Average value of the functionF8 Average value of the function
f (x)dxa
b
b a
Find
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F10 Value of y is increasing proportionally
to y
F10 Value ofF10 Value of .y is increasing y is increasing proportionally to yproportionally to y
dy
dtky
y Cekttranslating
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F11 Given , draw a
slope field dx
dy
F11 Draw a slope field of dy/dxF11 Draw a slope field of dy/dx
Using the given points and plug them into , drawing little lines with the indicated slopes at the points.
dx
dy
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G1 Given s(t) (position function), find v(t)
G1 Given position s(t), find v(t)G1 Given position s(t), find v(t)
Find v(t) = s’(t)
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G2 Given v(t) and s(0),
find s(t)
G2 Given v(t) and s(0), find s(t)G2 Given v(t) and s(0), find s(t)
s t v t dt C
P lu g in t = 0 to fin d C
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G4 Given v(t), determine if a particle is speeding up at
t = k
G4 Given v(t), determine if the particle G4 Given v(t), determine if the particle is speeding up at t=kis speeding up at t=k
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G7 Given velocity, v(t),
on [t1,t2], find the minimum acceleration of
the particle
G7 Given v(t), find minimum G7 Given v(t), find minimum accelerationacceleration
First find the acceleration
a(t)=v’(t) Then set a’(t) = 0 and minimize using a sign chart. Check critical values and t1, t2 to find the minimum.
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G8 Given the velocity function, find the average
velocity of a particle
on [a, b]
G8 Find the average rate of change G8 Find the average rate of change of velocity on [a,b]of velocity on [a,b]
v(t)dta
b
b a
s(b) s(a)
b a
Find
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G10 Given v(t), find how far a particle travels on
[a, b]
G10 Given v(t), find how far a particle G10 Given v(t), find how far a particle travels on [a,b]travels on [a,b]
v(t)a
b
dtFind
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G12 Given v(t) and s(0), find the greatest distance from the starting position of a particle on [0, t1]
G12 Given Given vv((tt)) and and ss(0)(0), find the greatest distance , find the greatest distance from the origin of a particle on [from the origin of a particle on [00, t, t11]]
Generate a sign chart of v(t) to find turning points.
Then integrate v(t) to get s(t), plug in s(0) to find the constant to c.
Finally, evaluate find s (t) at all turning points and find which one gives the maximum distance from your starting point, s(0).
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G15 Given a water tank with g gallons initially being filled at the rate of F(t) gallons/min and emptied at the rate of E(t) gallons/min on
1 2[ , ]t t
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G15a) the amount of water in
the tank at t = m minutes
G15a) Amount of water in the tank at t G15a) Amount of water in the tank at t minutesminutes
m
0
dt)]t(E)t(F[(g
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G15b) the rate the water
amount is changing
at t = m minutes
G15b) Rate the amount of water is changing at t = m
)m(E)m(Fdt)]t(E)t(F[(dt
d m
0
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G15c) the time when the
water is at a minimum
G15c) The time when the water is at a G15c) The time when the water is at a minimumminimum
Set F(m) - E(m)=0, solve for
m, and evaluate
at values of m AND endpoints
m
0
dt)]t(E)t(F[(g
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37. The rate of change of population is …
37 Rate of change37 Rate of change of a population of a population
dP
dt...
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62.Find
if
limx
f (x)
g(x)
limx
f (x) limx
g(x) 0
62 Find62 Find
Use l’Hopital’s Rule
limx
f (x)
g(x)
