When you see… A1. Find the zeros You think…. A1 To find the zeros

When you see…

A1. Find the zeros

You think…

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).

You think…

When you see…

A3 Show that f(x) is even

A3 Even functionA3 Even function

You think…

When you see…

A4 Show that f(x) is odd

A4 Odd functionA4 Odd function

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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).

You think…

When you see…

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

You think…

When you see…

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 . . . .

You think…

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)

You think…

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

You think…

When you see…

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)

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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)

You think…

When you see…

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

When you see…

C5 Find equation of the line tangent to f(x) at (x1,y1)

You think…

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

You think…

When you see…

C6 Find equation of the line normal to f(x) at (a, b)

C6 Equation of the normal lineC6 Equation of the normal line

You think…

When you see…

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

You think…

When you see…

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.

You think…

When you see…

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.

You think…

When you see…

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)

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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.

You think…

When you see…

D2 Find the interval(s) where f(x) is

increasing/dec.

f ’(x) < 0 means decreasing

D2 fD2 f(x) increasing(x) increasing

You think…

When you see…

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)

You think…

When you see…

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 + )

You think…

When you see…

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)

You think…

When you see…

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)

You think…

When you see…

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)

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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)

You think…

When you see…

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.

You think…

When you see…

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

You think…

When you see…

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)

You think…

When you see…

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]

You think…

When you see…

E2 Find area using right Riemann sums

E2 Area using right Riemann sumsE2 Area using right Riemann sums

A=base[x1+x2+x3…+xn]

You think…

When you see…

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.

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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[

You think…

When you see…

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

You think…

When you see…

E11

d

dxf (t)dt

a

x

E11 Fundamental TheoremE11 Fundamental Theorem

2nd FTC: Answer is f(x)

You think…

When you see…

d

dxf (t)dt

a

u

E12

E12 Fundamental Theorem, againE12 Fundamental Theorem, again

2nd FTC: Answer is

f (u)du

dx

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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 . . . .

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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)

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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.

You think…

When you see…

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

You think…

When you see…

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

You think…

When you see…

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).

When you see…

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

You think…

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

You think…

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

You think…

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

You think…

When you see…

37. The rate of change of population is …

37 Rate of change37 Rate of change of a population of a population

dP

dt...

You think…

When you see…

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)