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AP Calculus Review Notecards, with Example Questions -- Based on "When you see the words .... This is what you think of doing" Worksheet
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Terms in this set (79)
Find the zeros
aka: find the roots
set function = 0
* factor or use quadratic equation if quadratic
* graph to find 0s on calculator
Show that f(x) is even
Algebraically: show that f(-x)=f(x)
Graphically: see if symmetric about the y axis
Show that f(x) is odd
Algebraically: show that f(-x)= -f(x)
Graphically: see if symmetric about the origin
Show that lim x→a f(x) exists
Show that lim x→a- f(x) = lim x→a+ f(x)
Find lim x→a f(x), calculator allowed
Find values of the function for x-values close to a from BOTH the left AND the right
Find lim x→a f(x), no calculator
If you can substitute x=a directly in, then do it!
If you cannot substitute x=a in, try to:
* Factor/reduce
* Rationalize radicals
* Simplify complex fractions
* If piecewise, check if limit from the right = limit from the left
* use known trig limits
-- lim x→0 sinx/x = 1
-- lim x→0 (1-cosx)/x = 0
Find lim x→∞ f(x), calculator allowing
Do Y1(1000000) on your calculator
Find lim x→∞ f(x), no calculator
Think:
* small/big = 0
* big/small = DNE
* same/same = ratio of coefficients
same rules apply to x→-∞
Find horizontal asymptotes of f(x)
Find lim x→∞ f(x) and lim x→-∞ f(x)
Find vertical asymptotes of f(x)
Find where lim x->a f(x)=±∞
* factor/reduce f(x) and set denominator = 0
* remember ln x has VA at x=0
It's very important that you remember to factor FIRST before setting your denominator = 0. If a factor in the denominator crosses out with a factor in the numerator, then your function has a HOLE there and not a vertical asymptote.
Find domain of f(x)
Assume domain is (-∞,∞)
Restrictable domains:
*Denominators ≠ 0
*Square roots of only nonnegative numbers
*Log or ln of only positive numbers
*Real world constraints (ie: time≥0)
Show that f(x) is continuous
Show that 1=2=3:
1. lim x->a⁺ f(x)
2. lim x-> a⁻ f(x)
3. f(a)
Find the slope of the tangent line to f(x) at x=a
Find derivative f '(a) = m
Find equation of the line tangent to f(x) at (a,b)
f '(a) = m and use y-y₁=m(x-x₁) → y-b=m(x-a)
* if they don't give you the point and instead say "find the equation of the line tangent to f(x) at x=a", then we first find the point by computing f(a)
Find equation of the line normal (perpendicular) to f(x) at (a,b)
m = -1/f '(a) and use y-y₁=m(x-x₁) → y-b=m(x-a)
* if they don't give you the point and instead say "find the equation of the line normal to f(x) at x=a", then we first find the point by computing f(a)
Find average rate of change of f(x) on [a,b]
Algebra 2 brain! [f(b)-f(a)]/ [b-a]
Show that there exists a c in [a,b] such that f(c)=n
Intermediate Value Theorem (IVT)
Confirm f(x) is *continuous* on [a,b] then show that f(a) ≤ n≤ f(b)
Find the interval where f(x) is increasing
1. Find f '(x)
2. Find critical points (set both numerator and denominator to 0)
3. Make sign chart of f '(x)
4. Determine where f '(x) is positive
Find the interval where f(x) is decreasing
1. Find f '(x)
2. Find critical points (set both numerator and denominator to 0)
3. Make sign chart of f '(x)
4. Determine where f '(x) is negative
Find interval where the slope of f(x) is increasing
1. Find f ''(x)
2. Find critical points on f''(x) (set both numerator and denominator to 0)
3. Make sign chart of f ''(x)
4. Determine where f ''(x) is positive
Find instantaneous rate of change of f(x) at a
Find f '(a)
Given s(t) (position function), find v(t)
Find v(t) = s '(t)
Find f '(x) by the limit definition
(frequently asked backwards)
f '(a) = lim h→0 [f(a+h) -f(a)] / h
OR
f '(a)= lim x→a [f(x) -f(a)]/ x-a
Example:
Question1: lim h→0 [(2+h)³ - 8] / h
Answer1: f(x) = x³ and a=2 → f'(x) = 3x² → f'(2) = 3(2)²=12
Question2: lim x→9 [√(x) - 3] / [x-9]
Answer2: f(x) = √(x) and a=9 → f'(x) = 1/2 x^(-1/2) → f'(9) = 1/2 * (9)^(-1/2) = (1/2) * (1/3) = 1/6
Find the average velocity of a particle on [a,b]
Depending on if you know v(t) or s(t), use either formula.
Given v(t), determine if a particle is speeding up at t=k
Find v(k) and a(k)
If velocity and acceleration have the same sign at t=k, then the particle is speeding up.
If velocity and acceleration have opposite signs at t=k, then the particle is slowing down.
Given a graph of f '(x) find where f(x) is increasing
Determine where f '(x) is positive (above the x-axis)
Given a table of x and f(x) on selected values between a and b estimate f '(c) where c is between a and b
Straddle c using a value k greater than c and a value h less than c. Find the slope between k & c and the slope between c & h. Then average these two slopes to approximate f'(c).
Given a graph of f '(x), find where f(x) has a relative max
Find where:
* f '(x)=0 crosses the x-axis from positive values to negative values
* where f '(x) DNE and jumps from positive values to negative values
Given a graph of f'(x), find where f(x) is concave down
Identify where f '(x) is decreasing (because f concave down means f''(x)<0, which means f' is decreasing)
Given a graph of f'(x), find where f(x) has point(s) of inflection
Identify where f '(x) changes from increasing to decreasing or vice versa (because f has points of inflection when f changes concavity, aka f'' changes sign)
Show that a piecewise function is differentiable at the point a where the function rule splits
First, be sure function is *continuous* at x=a by evaulating each function at x=a.
Then, take the derivative of each piece and show that: lim x→a⁺ f '(x) = lim x→a⁻ f '(x)
Given the equation for f(x) and h(x)=f⁻¹(x) find h'(a)
Given the equation for f(x), find its derivative algebraically
1. know product/quotient/chain rules
2. know derivatives of basic functions
-- power rule: polynomials, radicals, rationals
-- e^x, b^x
-- lnx, logx
-- sinx, cosx, tanx
-- arcsinx (aka: sin⁻¹x), arccosx, arctanx
Find d²y/dx²
This is the same thing as asking for the second derivative, f ''(x).
Given a relation of x and y, find dy/dx algebraically
Implicit differentiation:
Find derivative of each term using product/quotient/chain appropriately:
1. Remember that every y has a "baby" (every derivative of y is multiplied by dy/dx aka y')
2. Then group all dy/dx terms on one side
3. Then factor out dy/dx and solve
Find derivative of f(g(x))
Chain rule
f '(g(x)) * g '(x)
Find the minimum value of function on [a,b]
1. Find local mins:
* find critical numbers of f (f '(x) =0 or f '(x) DNE)
* make sign chart
* find where sign change from negative to positive
2. Consider endpoints
3. Compare local min values to endpoint values and choose the smallest
Find the maximum slope of function on [a,b]
(maximum slope is asking us to find the maximum of f', so instead of finding max values of f like we usually do, just "take it down a level" and use the same procedure to minimize f')
1. find critical numbers of f' (f''(x)=0 or f''(x) DNE
* make sign chart
* find where sign change from positive to negative for local max
2. Consider endpoints by plugging back into f'(x)
3. Compare local min f'(x) values to endpoint f'(x) values and choose the smallest
Find critical values
Find where f'(x) = 0 or f'(x) DNE
Express f'(x) as a fraction and solve for numerator = 0 AND denominator = 0.
Find absolute max of f(x)
1. Find local mins:
* find critical numbers of f (f '(x) =0 or f '(x) DNE)
* make sign chart
* find sign change from positive to negative
2. Compare local mins and find the smallest value
3. See if you can use the "for all" tip (ie: f is increasing for all x<2 and f is decreasing for all x>2, therefore x=2 is an absolute max)
Show that there exists a c in [a,b] such that f'(c)=0
Rolle's Theorem
Confirm that f is *continuous AND differentiable* on the interval
Find k and j in [a,b] such that f(k)=f(j), then there is some c in [k,j] such that f'(c)=0
Examples: http://www.shmoop.com/derivatives/rolles-theorem-examples.html
Show that there exists a c in [a,b] such that f '(c)=m
Mean Value Theorem
Confirm that f is *continuous AND differentiable* on the interval
Find k and j in [a,b] such that m = [f(k)-f(j)] / [k-j], then there is some c in [k,j] such that f '(c)=m
*think: slope = derivative somewhere*
Examples: http://www.sosmath.com/calculus/diff/der11/der11.html
Find range of f(x) on [a,b]
Use max/min techniques to find values at relative max/mins. Also compare to endpoints f(a) and f(b).
This is the same as finding the absolute max/min of the function on a closed interval. (See notecard #40 - find the absolute max of f(x).) The absolute min VALUE will be the lower bound of the range and the absolute max VALUE will be the upper bound of the range.
Find range of f(x) on (-∞,∞)
Use max/min techniques to find values at relative max/mins. Also compare to lim x→±∞ f(x)
Find the locations of relative extrema of f(x) given both f'(x) and f ''(x)
*This is particularly useful when using the number line method & finding a sign change in f'(x) is difficult*
2nd derivative test
1. Find where f '(x)=0 or DNE
2. Check the value of f''(x) for those values identified in step 1
* If f'(x)=0 and f''(x)>0 (indicating CU), then we have a local MIN
* If f'(x)=0 and f''(x)<0 (indicating CD), then we have a local MAX
Find inflection points of f(x) algebraically
Express f ''(x) as a fraction and set both numerator=0 and denominator=0
Make sign chart of f ''(x) to find where f''(x) changes sign
*note: confirm that f(x) exists for any x values that make f ''(x) DNE!*
Show that the line y=mx+b is tangent to f(x) at (x₁, y₁)
2 relationships required: same slope and point of intersection
1. check that m=f'(x₁)
2. confirm that (x₁, y₁) is on both f(x) and the tangent line
Find any horizontal tangent line(s) to f(x)
1. write dy/dx as a fraction
2. set numerator=0
* NOTE: confirm values are on the curve by plugging them back in*
Approximate the value of f(0.1) by using the tangent line to f at x=0
Find the equation of the tangent line at x=0:
* Find slope: m = f'(0)
* Find point: (0, f(0))
* Find equation to tangent line using: y-y₁=m(x-x₁)
Using the equation found above, approximate f(0.1) by plugging 0.1 in for x.
Find rates of change for volume problems
Write volume formula
Find dV/dt
*Be careful about product/chain rules
*Watch positive (if the measurement/rate is increasing) vs negative (if the measurement/rate is decreasing) signs for rates
Find any vertical tangent line(s) to f(x)
1. write dy/dx as a fraction
2. set denominator=0
* NOTE: confirm values are on the curve by plugging them back in *
Find rates of change for pythagorean theorem problems
x²+y²=z²
2x (dx/dt) + 2y (dy/dt) = 2z (dz/dt)
Reduce by dividing everything above by 2
*Watch positive (if the measurement/rate is increasing) vs negative (if the measurement/rate is decreasing) signs for rates
Find the average value of f(x) on [a,b]
vs Find the average rate of change of f(x) on [a,b]
Given v(t), find the total distance a particle travels
find ∫ |v(t)| dt
example: https://www.youtube.com/watch?v=sT6KedSyfiE&feature=youtu.be&t=12m25s
Given v(t), find the change in position of a particle
find ∫ v(t) dt
example: https://www.youtube.com/watch?v=sT6KedSyfiE&feature=youtu.be&t=12m25s
Given v(t) and initial position of a particle, find the position at t=a
d/dx ∫ from a to x of f(t)dt
d/dx ∫ from a to g(x) of f(t)dt
Find area using left riemann sums
A = width * [f(x₀) + width * f(x₁) + ... + width * f(xn-₁)]
(where n is the number of rectangles)
NOTE: sketch a number line to visualize & choose LEFT endpoints
Find area using right riemann sums
A = width * [f(x₁) + width * f(x₂) + ... + width * f(xn)]
(where n is the number of rectangles)
NOTE: sketch a number line to visualize & choose RIGHT endpoints
Find area using midpoint rectangles
Typically done with table of values
*be sure to use only values that are given
-so if given 6 data points, can only do 3 midpoint rectangles!!!
NOTE: sketch a number line to visualize
Find area using trapezoids
Remember: height = average of the bases
A = width * [f(x₀)+f(x₁)]/2 + width * [f(x₁)+f(x₂)]/2 + ... + width * [f(xn-₁)+f(xn)]/2
Describe how you can tell if rectangle or trapezoid approximations over or under estimate area
DRAW A PICTURE and sketch two rectangles/trapezoids and you will clearly see if it is an overestimate or underestimate
given ∫ from a to b of f(x) dx,
find ∫ from a to b of [f(x)+k]dx
Given dy/dx, draw a slope field
Use given points and plug them into dy/dx, drawing little lines with indicated slopes at the points
y is increasing proportionally to y
dy/dt=ky translating to y=Ae^(kt)
Solve the differential equation
SEPARATE VARIABLES
* get all the x's & dx on one side, y's & dy on other
* integrate each side, add C to the right
* solve for y
* find value of C
when solving for y choose +OR- if necessary
Find volume given a base bounded by f(x) and g(x) with f(x)>g(x) and cross sections perpendicular to the x-axis are squares
the distance between curves is the base of your square
volume is ∫ [f(x)-g(x)]² dx (bounds are intersection points)
Given value of F(a) and F'(x)=f(x), find F(b)
Meaning of ∫ from a to b of f(t)dt
Accumulation function: net amount of y units for function f(x) beginning at x=a and ending at x=b
Given v(t) and s(0), find greatest distance from origin of a particle on [a,b]
1. Find maximum distance from origin, which happens when s'(t) = v(t) = 0 or v(t) DNE
Let the time we find = a
2. s(a) = s(0) + ∫ from 0 to a [v(t)] dt
*because ∫ from 0 to a [v(t)]dt = [s(t)] from 0 to a = s(a)-s(0) --> s(a) = s(a) + ∫ from 0 to a [v(t)] dt
3. Compare the y-value of each candidate with the y-value of each endpoint. Choose greatest distance (it might be negative!)
See question 2(d) http://apcentral.collegeboard.com/apc/members/repository/ap03_sg_calculus_ab_26472.pdf
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 [0,b] find...
-the amount of water in the tank at m minutes
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 [0,b] find...
-the rate the water amount is changing at m
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 [0,b] find...
-the time when the water is at a minimum
Solve F(t)-E(t)=0 to find candidates, evaluate candidates and endpoints. Choose min value.
Find the area between f(x) and g(x) with f(x)>g(x) on [a,b]
A=∫ a to b of [f(x)-g(x)] dx
Find the volume of the area between f(x) and g(x) with f(x)>g(x) rotated about the x axis
V= π ∫ a to b of [(f(x))²-(g(x))²]dx
Given v(t) and s(0), find s(t)
Find the line x=c that divides the area under f(x) on [a,b] to 2 equal areas
Find the volume given a base bounded by f(x) and g(x) with f(x)>g(x) and cross sections perpendicular to the x axis are semicircles
Distance between curves is DIAMETER of your circle
so volume:
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