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AP Calculus Exam Prep
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Gravity
Terms in this set (69)
Find the average velocity given the velocity function, v(t)
1/(b-a) ∫ v(t) dt
Find the average velocity given the position function, s(t)
[s(b) - s(a)] / [b-a]
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 allowed
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
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 v(t), determine where the particple is speeding up
Find intervals where the velocity function is moving *away* from the x-axis.
Justification: v(t) and a(t) have the *same sign* on this interval
Given a graph of f '(x) find where f(x) is increasing
Determine where f '(x) is positive (above the x-axis)
Given values for f(1), f(3), and f(5), approximate f'(3).
1. find the slope between x=1 & x=3 --> [f(3)-f(1)] / [3-1]
2. find the slope between x=3 & x=5 --> [f(5)-f(3)] / [5-3]
3. average the two slopes found above to approximate f'(3)
Given values for f(1), f(3), and f(5), approximate f'(2).
f'(2) is appoximated by finding the slope between x=1 & x=3 --> [f(3)-f(1)] / [3-1]
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)
d/dx sin^-1 (x)
1 / √(1-x²)
d/dx cos^-1 (x)
-1 / √(1-x²)
d/dx tan^-1 (x)
1 / [1+x²]
d/dx csc(x)
-cscx cotx
d/dx sec(x)
secx tanx
d/dx tan(x)
sec²x
Find d²y/dx²
This is the same thing as asking for the second derivative, f ''(x).
Find the derivative:
(y+1)(x²+5x)
Using implicit differentiation & product rule:
(dy/dx) (x²+5x) + (y+1)(2x+5)
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]
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.
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 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!*
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*
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 *
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 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 s(2), find s(5)
s(5) = s(2) + ∫₂⁵ v(t) dt
d/dx ∫ from a to x of f(t)dt
f(x) (by FTC)
d/dx ∫ from a to 5x³ of f(t)dt
f(5x³) * 15x² (by FTC & Chain)
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
If a function is increasing & concave up, then: how you can tell if rectangle or trapezoid approximations over or under estimate area
Trapezoids = overapproximation
Left endpoints = underapproximation
Right endpoints = overapproximation
If a function is increasing & concave down, then: how you can tell if rectangle or trapezoid approximations over or under estimate area
Trapezoids = underapproximation
Left endpoints = underapproximation
Right endpoints = overapproximation
If a function is decreasing & concave down, then: how you can tell if rectangle or trapezoid approximations over or under estimate area
Trapezoids = underapproximation
Left endpoints = overapproximation
Right endpoints = underapproximation
given ∫ from a to b of f(x) dx,
find ∫ from a to b of [f(x)+k]dx
y is increasing proportionally to y
dy/dt=ky translating to y=Ae^(kt)
Solve the differential equation, find the general solution
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 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
g + ∫ from 0 to m F(t)-E(t) dt
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.
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
∫ from a to c of f(x) dx = ∫ from c to b of f(x) dx
d/dx a^x
a^x ln(a)
d/dx log base a of x
1 / [xlna]
If lim x→a f(x)/g(x) = 0/0 or ∞/∞
Use l'Hopital!
∴ lim x→a f(x)/g(x)
= lim x→a f'(x)/g'(x)
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