AP Calculus BC Final QUESTIONS &
ANSWERS(RATED A+)
taking a continuous object and breaking it up to approximate (advantages,
disadvantages) - ANSWERdisadvantage- more pieces to deal with
advantage- more accurate estimate
Calculus - ANSWERall about limits and the attempt of closing in on an exact value of
an answer by using finer and finer approximations
limit - ANSWERthe idea of a limit is that we are attempting to see what the output (y)
approaches, if anything, as the input (x) approaches a particular value
limit (L) can be one of four things - ANSWER-a number (if f(x) closes in on a
number)
-∞ (if f(x) goes up forever without approaching a horizontal asymptote)
- - ∞ (if f(x) goes down forever without approaching a horizontal asymptote)
-does not exist (if f(x) doesn't stabilize or imaginary or jumps)
three ways to determine limits - ANSWER-graphically (examine the plot)
-numerically (substitute closer and closer values into x with calculator)
-algebraically (manipulating equations into a simpler form and sometimes use direct
substitution)
determine limits algebraically - ANSWERlook at what's happening to the y-
coordinate as you close in on a particular x-value
average speed - ANSWERdistance traveled/ change in time
average rate of change- "average" - ANSWERa measure of what the "central" value
is at all our speed/rates at a given interval
secant line
instantaneous rate of change- "instantaneous" - ANSWERa measure of what the
value is of your speed/rate at a given moment in time
tangent line
two-sided limits - ANSWERfunction needs to be approaching the same y-value from
both the left and right to exist
one-sided limits - ANSWERonly focus on the y-value from one particular side
determine limits numerically - ANSWERsubstitute in numbers closer and closer to "c"
and watching what happens to L
Determining Limits Algebraically (when x is not heading to infinite or - infinite) -
ANSWERbreak single large limit into several smaller easier limits
, Sandwich or Squeeze Theorem - ANSWERif you are trying to determine the limit of
f(x) and can find a function g(x) ≤ f(x) and another function h(x) ≥ f(x), if g(x) and h(x)
have the same limit as c, then f(x) must also have the same limit as c
rational function - ANSWERpolynomial/polynomial
degree - ANSWERhighest power x is raised to in a polynomial
leading coefficient - ANSWERcoefficient in front of the highest power of x
# horizontal asymptotes for rational functions - ANSWER1
graphs get interrupted by discontinuities - ANSWERhole, single-point jump,
piecewise equations jumps, vertical asymptotes
removeable discontinuity - ANSWERif you could redefine just one point, can turn into
function with no breaks
non-removeable discontinuity - ANSWERlimit at discontinuity is either infinite, -
infinite, or DNE→ non-removable- either jump discontinuities (limit does not exist) or
infinite discontinuities (limit is infinite or - infinite)
Intermediate Value Theorem (for continuous functions) - ANSWERif f(x) is
continuous everywhere on [a,b] then the graph visits every y-value from f(a) to f(b) at
least once
velocity - ANSWERvector (direction and magnitude)
speed - ANSWERscalar (only magnitude)
Strategy for drawing f'(x) from f(x) - ANSWER1. find where f(x) has a slope of zero
2. bracket where f(x) is positive and negative slope
3. find the actual positive and negative slopes if possible
4. graph slope f'(x) v. t (Note: make x-axis identical for both graphs)
sometimes you don't have enough information to find exact slope. if so, come up with
some estimates and notice if getting steeper or shallower
differentiability - ANSWERlimit of the slope on both sides of every point must be the
same finite value and it must be continuous
1. continuity is required for differentiability
2. continuity does not guarantee differentiability
3. differentiability does not guarantee continuity
numerical derivative - ANSWERnumerical value for the slope of the tangent line
actual derivative - ANSWERapproach separately from both sides and make sure if
the limits of the slope match
ANSWERS(RATED A+)
taking a continuous object and breaking it up to approximate (advantages,
disadvantages) - ANSWERdisadvantage- more pieces to deal with
advantage- more accurate estimate
Calculus - ANSWERall about limits and the attempt of closing in on an exact value of
an answer by using finer and finer approximations
limit - ANSWERthe idea of a limit is that we are attempting to see what the output (y)
approaches, if anything, as the input (x) approaches a particular value
limit (L) can be one of four things - ANSWER-a number (if f(x) closes in on a
number)
-∞ (if f(x) goes up forever without approaching a horizontal asymptote)
- - ∞ (if f(x) goes down forever without approaching a horizontal asymptote)
-does not exist (if f(x) doesn't stabilize or imaginary or jumps)
three ways to determine limits - ANSWER-graphically (examine the plot)
-numerically (substitute closer and closer values into x with calculator)
-algebraically (manipulating equations into a simpler form and sometimes use direct
substitution)
determine limits algebraically - ANSWERlook at what's happening to the y-
coordinate as you close in on a particular x-value
average speed - ANSWERdistance traveled/ change in time
average rate of change- "average" - ANSWERa measure of what the "central" value
is at all our speed/rates at a given interval
secant line
instantaneous rate of change- "instantaneous" - ANSWERa measure of what the
value is of your speed/rate at a given moment in time
tangent line
two-sided limits - ANSWERfunction needs to be approaching the same y-value from
both the left and right to exist
one-sided limits - ANSWERonly focus on the y-value from one particular side
determine limits numerically - ANSWERsubstitute in numbers closer and closer to "c"
and watching what happens to L
Determining Limits Algebraically (when x is not heading to infinite or - infinite) -
ANSWERbreak single large limit into several smaller easier limits
, Sandwich or Squeeze Theorem - ANSWERif you are trying to determine the limit of
f(x) and can find a function g(x) ≤ f(x) and another function h(x) ≥ f(x), if g(x) and h(x)
have the same limit as c, then f(x) must also have the same limit as c
rational function - ANSWERpolynomial/polynomial
degree - ANSWERhighest power x is raised to in a polynomial
leading coefficient - ANSWERcoefficient in front of the highest power of x
# horizontal asymptotes for rational functions - ANSWER1
graphs get interrupted by discontinuities - ANSWERhole, single-point jump,
piecewise equations jumps, vertical asymptotes
removeable discontinuity - ANSWERif you could redefine just one point, can turn into
function with no breaks
non-removeable discontinuity - ANSWERlimit at discontinuity is either infinite, -
infinite, or DNE→ non-removable- either jump discontinuities (limit does not exist) or
infinite discontinuities (limit is infinite or - infinite)
Intermediate Value Theorem (for continuous functions) - ANSWERif f(x) is
continuous everywhere on [a,b] then the graph visits every y-value from f(a) to f(b) at
least once
velocity - ANSWERvector (direction and magnitude)
speed - ANSWERscalar (only magnitude)
Strategy for drawing f'(x) from f(x) - ANSWER1. find where f(x) has a slope of zero
2. bracket where f(x) is positive and negative slope
3. find the actual positive and negative slopes if possible
4. graph slope f'(x) v. t (Note: make x-axis identical for both graphs)
sometimes you don't have enough information to find exact slope. if so, come up with
some estimates and notice if getting steeper or shallower
differentiability - ANSWERlimit of the slope on both sides of every point must be the
same finite value and it must be continuous
1. continuity is required for differentiability
2. continuity does not guarantee differentiability
3. differentiability does not guarantee continuity
numerical derivative - ANSWERnumerical value for the slope of the tangent line
actual derivative - ANSWERapproach separately from both sides and make sure if
the limits of the slope match
