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59 Cards in this Set
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In what situation would using synthetic division for finding limits work?

When the numerator is at least two degrees smaller than the denominator; e.g., f(x) = (x^31)/(x1)


Explain how you could find a limit (without a calculator) of a function with a numerator like √ (x+5) +3?

Multiply by √ (x+5)  3 over √ (x+5)  3 so that the radicals cancel!!


What is the limit as x approaches zero of (sin4x)/x ?

4


What is the limit as x approaches zero of (sin2x)/(sin3x) ?

2/3


What is the limit as x approaches zero of (tanx)^2 / x

0


The definition of continuity at a graph says that f is continuous at x = c if.... [3]

1. f(c) exists at x=c
2. The limit as x approaches c of f(x) exists. 3. f(c) = limit as x approaches c 

What should you end your continuity theorem with?

∴ f(x) is/not continuous at x =c


List the steps of continuity for a closed interval [a,b].

1. (sweeping statement) "f(x) is cont. on open interval (a,b) because it meets all 3 tests of continuity"
2. f(a) exists, f(b) exists 3. limit as x approaches a (+) exists; limit as x approaches b () exists. 4. f(a) = limit as x approaches a (+) f(b) = limit as x approaches b () 

What should you end the closed interval continuity theorem with?

∴ f(x) is continuous/not on [a,b]


State the IVT (intermediate value theorem).

If f(x) is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then at least one such value "c" exists such that f(c)=k on [a,b].


E.g., if f(x) = (x^2+x)/(x1), show that x=c exists such in [2.5,4] such that f(c)=6.

1. f(x) is cont. on [a,b]
2. f(2.5) = 5.833 f(4) = 6.667 f(c) = k = 6 3. f(2.5) ≤ f(c) ≤ f(4) or 5.883 ≤ 6 ≤ 6.667 

What should you end your IVT proof with?

∴ by the IVT, there exists one such value c on [2.5,4] such that f(c) = 6.


What will direct substitution of a limit yield when there's a vertical asymptote? What can you do to prove it?

It will yield a nonzero number over zero; to prove it's a V.A., take the limit of that number from the left () and the right (+). One should equal ∞, the other ∞.


What is the original position function of a real object in ft?

s(t) = 16x^2 bx + c


Describe the meanings of b and c in the position function: s(t) = 16x^2 bx + c

b = initial velocity (zero if dropped) (ft/s)
c = initial height (ft) 

What is the original position function of a real object in m?

s(t) = 4.9x^2 bx + c


Describe the meanings of b and c in the position function: s(t) = 4.9x^2 bx + c

b = initial velocity (zero if dropped) (m/s)
c = initial height (m) 

What is the difference between the tangent and normal line at the same point?

Tangent is found by regular methods; normal is the opposite reciprocal of this


What should you do if asked to find the tangent line of an equation such as x^2 + y^2 = 9?

Use implicit differentiation and once dy/dx is found, plug in the given point for x and y.


What should you do to find the points of horizontal tangency of an equation requiring implicit differentiation?

Set dy/dx to zero (denominator cancels out) and solve for the numerator. Then, plug this answer back into the original equation to yield both coordinates of the point.


What should you do to find the points of vertical tangency of an equation requiring implicit differentiation?

This occurs were dy/dx DNE, so set the denominator of the dy/dx to zero, and solve for this variable. Then make sure to plug this answer back into the original to yield both coordinates of the point.


1. Identify the equation to be used
2. Identify the "given" and "trying to find" 3. d/dt the entire equation 4. Plug and chug and don't freeze too soon ...are all the steps of... 
Related rates problems


What does the Extreme Value Theorem state?

If f is cont. on closed interval [a,b], then f must have a maximum and a minimum.


What should you do when asked to find the absolute max and min of a function?

Derive it to find the critical numbers, THEN plug the CNs AND the interval numbers into the original function. Which ever yields the smallest and biggest are the mins and maxs, respectively.


State Rolle's theorem. (Remember, it's different because it addresses the derivative.)

If f is cont on [a,b] and differentiable on (a,b), and if f(a) = f(b), then there must exist one value "c" on OPEN INTERVAL (a,b) such that f'(c) = 0.


T/F: When actually finding this "c" value, you find the derivative, set it to zero, and solve for x. But, you must make sure to only include the c values that are in the OPEN interval. The others are not relevant.

True


How can you distinguish the Mean Value Theorem (for derivatives) from the other previous theorems?

It addresses the derivative as well but it actually sets the derivative to the found algebra slope, not to zero.


State the Mean Value Theorem for derivatives.

If f is cont on [a,b] and differentiable on (a,b), then there exists at least one value "c" on (a,b) such that the algebra slope ((f(b)f(a))/ba) equals the calculus slope (f'(c)).


When using the MVT, describe what should be written down and the steps.

1. cont ✓
2. diff ✓ ∴ MVT applies Then find algebra slope using regular formula, set the derivative equal to this, and find the x=c values within the given open interval. 

When f''(CN) < 0, what is it?

A relative maximum!! (remember the smiley)


When f''(CN) > 0, what is it?

A relative minimum! (smiley)


At infinity, what is the limit of a constant over a polynomial?

zero


At infinity, what is the limit of a poly^r/polr^q, such that r<q?

zero


At infinity, what is the limit of a poly^r/polr^q, such that r>q?

infinity, or negative infinity if that is the asked limit


At infinity, what is the limit of a poly^r/polr^q, such that r=q?

The leading coefficient over the leading coefficient.


Again, if r<q, where is the asymptote?

At y=0


Again, if r=q, where is the asymptote?

The leading coefficient over the leading coefficient.


Where is the asymptote of r = q +1 (thus the degree of the num. is 1 greater than the degree of the denom.)?

An Oblique asymptote is here at y=mx+b, found by synthetic division


If f(x) = f(x), what type of symmetry is there?

yaxis symmetry


If f(x) = f(x), what type of symmetry is there?

Origin symmetry


Describe the two equations in optimization.

1° (primary)  the equation being maximized/minimized
2° (secondary)  the fixed equation 

Describe the process of optimization with the 1° and 2° equations.

Isolate a variable in the fixed 2° equation and replace it in the 1° equation. Then, derive the 1° equation and find the CN. Plug this CN back into the isolated 2° equation if necessary.


What is the distance formula?

√(x1x2)^2 + (y1y2)^2


Describe the process you should take when given a point and asked to find the point on f(x) closest to it.

Plug the given point into the distance formula and derive this without the radical. Once you derive it, set it to zero to find the CN. Plug the CN back into the original equation to fine the y coordinate, and this is your point.


If asked to find the ACTUAL change in y, what formula should be used?

Δy = f(x + Δx)  f(x)


If asked to find the ESTIMATED change in y, what formula should be used?

dy = f'(x)dx


How can you use the tangent line to estimate a value like √16.5?

Use the point (16, 4) and .5 as Δx. Use f(x) = √x and find f'(16) to yield the slope. Use the slope and point to find the line and finally plug in 16.5.


If also estimating a function value, what equation can be used?

f(x + Δx) ≈ f(x) + f'(x)dx


Explain how to calculate the propagated error, if a square is measured to be 12 inches with ±(1/16) an inch of error.

x = 12, Δx = 1/16, A = x^2
dA/dx = 2x dA = 2xdx = 2(12)(1/16) = 1.5 

For the last problem, how would PERCENT error be calculated?

dA/A = 2xdx/x^2 = 2dx/x = 2(1/16)/12 x 100 = 1.04%


T/F: If asked for the ALLOWABLE percent error for 2.5 error propagation, you simply set the dA/A to less than or equal to .025 and solve for dx/x. (Multiply this by 100) and this will yield the allowable percent error.

True


If given an initial height and velocity of a ball function, what will yield its maximum height?

Find the CNs of the velocity function (remember s(t) = 4.9x^2 bx + c, then derive to find velocity) and this will be the maximum at t. Plug this into the position function to find the actual height.


What does the MVT for integrals say?

There is a point "c" on [a, b] where we can draw rectangular area to equal integral area. (review the formula, simple)


When given a velocity function and asked to find the total distance traveled on a certain interval, what do you do?

Take the integral from a to b of the absolute value of that velocity function.


If asked to find the integral of a function from x to x+2 instead of from a simple a to b, what should you do?

Split it up into two integrals from x to a, and then from a to x+2. Then, put a negative in front of the first one to make it from a to x, and then use the FTC.


T/F: **REMEMBER that when using u substitution, the interval of the integral should be from a to b, not the original specific values. If you choose not to go back to x world (sometimes you have to stay), use the "Let u = x" equation to solve for you and use these as your intervals.

True


T/F: In integration, there is a "trick" in which you add a plus one (or any other number) to the numerator of the integral so that it becomes factorable and cancels out with the denominator. Then, the "minus one" must be added and integrated with a separate integral.

True


When does a function have an inverse?

Only when it passes the vertical AND horizontal line test, and thus it is one to one.


State the relationship between a function, its inverse, and its derivativeinverse.

If f(a)=b, then f1(b)=a.
f'(a)= 1/f1'(b) 