--Ap Calculus Exam Notes

Front 1

In what situation would using synthetic division for finding limits work?

Back 1

When the numerator is at least two degrees smaller than the denominator; e.g., f(x) = (x^3-1)/(x-1)

Front 2

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

Back 2

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

Front 3

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

Back 3

4

Front 4

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

Back 4

2/3

Front 5

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

Back 5

0

Front 6

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

Back 6

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

Front 7

What should you end your continuity theorem with?

Back 7

∴ f(x) is/not continuous at x =c

Front 8

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

Back 8

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 (-)

Front 9

What should you end the closed interval continuity theorem with?

Back 9

∴ f(x) is continuous/not on [a,b]

Front 10

State the IVT (intermediate value theorem).

Back 10

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].

Front 11

E.g., if f(x) = (x^2+x)/(x-1), show that x=c exists such in [2.5,4] such that f(c)=6.

Back 11

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

Front 12

What should you end your IVT proof with?

Back 12

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

Front 13

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

Back 13

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 -∞.

Front 14

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

Back 14

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

Front 15

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

Back 15

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

Front 16

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

Back 16

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

Front 17

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

Back 17

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

Front 18

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

Back 18

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

Front 19

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

Back 19

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

Front 20

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

Back 20

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.

Front 21

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

Back 21

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.

Front 22

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...

Back 22

Related rates problems

Front 23

What does the Extreme Value Theorem state?

Back 23

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

Front 24

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

Back 24

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.

Front 25

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

Back 25

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.

Front 26

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.

Back 26

True

Front 27

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

Back 27

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

Front 28

State the Mean Value Theorem for derivatives.

Back 28

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))/b-a) equals the calculus slope (f'(c)).

Front 29

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

Back 29

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.

Front 30

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

Back 30

A relative maximum!! (remember the smiley)

Front 31

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

Back 31

A relative minimum! (smiley)

Front 32

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

Back 32

zero

Front 33

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

Back 33

zero

Front 34

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

Back 34

infinity, or negative infinity if that is the asked limit

Front 35

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

Back 35

The leading coefficient over the leading coefficient.

Front 36

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

Back 36

At y=0

Front 37

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

Back 37

The leading coefficient over the leading coefficient.

Front 38

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

Back 38

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

Front 39

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

Back 39

y-axis symmetry

Front 40

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

Back 40

Origin symmetry

Front 41

Describe the two equations in optimization.

Back 41

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

Front 42

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

Back 42

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.

Front 43

What is the distance formula?

Back 43

√(x1-x2)^2 + (y1-y2)^2

Front 44

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

Back 44

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.

Front 45

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

Back 45

Δy = f(x + Δx) - f(x)

Front 46

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

Back 46

dy = f'(x)dx

Front 47

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

Back 47

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.

Front 48

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

Back 48

f(x + Δx) ≈ f(x) + f'(x)dx

Front 49

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

Back 49

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

Front 50

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

Back 50

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

Front 51

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.

Back 51

True

Front 52

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

Back 52

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.

Front 53

What does the MVT for integrals say?

Back 53

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

Front 54

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

Back 54

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

Front 55

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?

Back 55

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.

Front 56

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.

Back 56

True

Front 57

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.

Back 57

True

Front 58

When does a function have an inverse?

Back 58

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

Front 59

State the relationship between a function, its inverse, and its derivative-inverse.

Back 59

If f(a)=b, then f-1(b)=a.

f'(a)= 1/f-1'(b)