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

A1 To find the zeros...

When you see… A2. Find intersection of f(x) and g(x) You think…

A2 To find the zeros...

You think… When you see… A3 Show that f(x) is even

A3 Even function

You think… When you see… A4 Show that f(x) is odd

A4 Odd function

You think… When you see… 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) 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, 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<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

B1 Find B1 Find 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)

You think… When you see… B2 Find where f(x) is a piecewise function.

B2 Show exists (Piecewise) Check 1-sided limits....

You think… B3 When you see… Show that f(x) is continuous

. B3 f(x) is continuous 1) exists 2) exists 3)

You think… B4 When you see… Find

B4 Find Express f(x) as a fraction, determine highest power. If in denominator, limit = 0 If in numerator, lim = +

You think… When you see… B-5 Find horizontal asymptotes of f(x)

Findand B5 Find horizontal asymptotes of f(x)

You think… When you see… C1 Find f ’(x) by definition

C1Find f ‘( x) by definition

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) 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) 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.

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

When you see… C5 Find equation of the line tangent to f(x) at (x 1,y 1 ) You think…

C5 Equation of the tangent line Find slope m = f ’(x). Use point (x 1, y 1 ) Use Point Slope Equation: y – y 1 = x – x 1

You think… When you see… C6 Find equation of the line normal to f(x) at (a, b)

C6 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 line

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) Write f ’(x) as a fraction. Set the denominator equal to zero.

You think… When you see… C9 Approximate the value of f (x 1 + a) if you know the function goes through (x 1, y 1 )

C9 C9 Approximate the value of (x 1 + a) Find the equation of the tangent line to f using y-y 1 = m(x-x 1 ). Now evaluate at x = x 1 +a. Note: The closer to a is to x 1, 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 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 (x 1, y 1 )

y = mx+b is tangent to f(x) at (a,b) C11 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 x 1 : ( m=f ’(x) ) 2)The function and line have same y-value at x 1

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 f(x) at x=a On g use (a, Q) On f use (Q, a) Find Q-value So

C12 Derivative of g(x), the inverse of f(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

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 differentiable at x=a 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 values

You think… When you see… D2 Find the interval(s) where f(x) is increasing/dec.

f ’(x) < 0 means decreasing D2 f(x) increasing

You think… When you see… D3 Find points of relative extrema of f(x)

D3 Find relative extrema

You think… When you see… D4 Find inflection points

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) 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) on Use max/min techniques to find relative max/mins Then examine

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] 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]

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] 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

D10 Given a graph of f ‘(x), find where f(x) is 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(x 1 +a) is over- or underestimate of actual f(x 1 + a)

D11 Determine whether f(x 1 + a) is over- or underestimate See C9 above. Find f(x 1 +a). Find f ″ on an interval containing x 1. 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 f (x) is increasing

You think… When you see… D13 Find the minimum slope of a function

D13 Minimum slope of a function

You think… When you see… E1 Find area using left Riemann sums

E1 Area using left Riemann sums A=base[x 0 +x 1 +x 2 …+x n-1 ]

You think… When you see… E2 Find area using right Riemann sums

E2 Area using right Riemann sums A=base[x 1 +x 2 +x 3 …+x n ]

You think… When you see… E3 Find area using midpoint rectangles

E3 Area using midpoint rectangles

You think… When you see… E4 Find area using trapezoids

E4 Area using trapezoids This formula only works when the base is the same. If not, you must do individual trapezoids. I would EXPECT this!

You think… When you see… E8 Meaning of

E8 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

E9 Given area under a curve and vertical shift, find the new area under the curve

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 F(a) and the that the anti-derivative of f is F, find F(b) 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.

You think… When you see… E11

E11 Fundamental Theorem 2 nd FTC: Answer is f(x)

You think… When you see… E12

E12 Fundamental Theorem, again 2nd FTC: Answer is

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] Assuming that the f curve is above the g curve

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) is divided equally

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 between f(x) and g(x) about the x-axis Assuming that the f curve is above the g curve

You think… When you see… F6 Given a base, cross sections perpendicular to the x-axis that are squares

F6 Square cross sections perpendicular to the x-axis The area between the curves is typically the base of the square so the volume is

You think… When you see… F7 Solve the differential equation …

F7 Solve the differential equation...

You think… When you see… F8 Find the average value of f(x ) on [a,b]

F8 Average value of the function Find

You think… When you see… F10 Value of y is increasing proportionally to y

F10 Value ofy is increasing proportionally to y F10 Value of. y is increasing proportionally to y translating

You think… When you see… F11 Given, draw a slope field

F11 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.

You think… When you see… G1 Given s(t) (position function), 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)

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 is speeding up at t=k

You think… When you see… G7 Given velocity, v( t ), on [ t 1,t 2 ], find the minimum acceleration of the particle

G7 Given v(t), find minimum acceleration First find the acceleration a(t)=v’(t) Then set a’(t) = 0 and minimize using a sign chart. Check critical values and t 1, t 2 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 of velocity on [a,b] Find

You think… When you see… G10 Given v(t), find how far a particle travels on [a, b]

Find

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, t 1 ]

Given v(t) and s(0), find the greatest distance from the origin of a particle on [ 0, t 1 ] G12 Given v(t) and s(0), find the greatest distance from the origin of a particle on [ 0, t 1 ] 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

You think… G15a) the amount of water in the tank at t = m minutes

G15a) Amount of water in the tank at t minutes

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

You think… G15c) the time when the water is at a minimum

G15c) The time when the water is at a minimum Set F(m) - E(m)=0, solve for m, and evaluate at values of m AND endpoints

You think… When you see… 37. The rate of change of population is …

37 Rate of change of a population

You think… When you see… 62.Find if

62 Find Use l’Hopital’s Rule