AP Calculus BC Final Exam Questions and Answers
taking a continuous object and breaking it up to approximate (advantages,
disadvantages) - Answer-disadvantage- more pieces to deal with
advantage- more accurate estimate
/.Calculus - Answer-all about limits and the attempt of closing in on an exact value of an
answer by using finer and finer approximations
/.limit - Answer-o the 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 - Answer-look at what's happening to the y-coordinate as
you close in on a particular x-value
/.two-sided limits - Answer-function needs to be approaching the same y-value from
both the left and right to exist
/.one-sided limits - Answer-only focus on the y-value from one particular side
/.determine limits numerically - Answer-substitute 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) -
Answer-break single large limit into several smaller easier limits
/.Sandwich or Squeeze Theorem - Answer-if 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 - Answer-polynomial/polynomial
/.degree - Answer-highest power x is raised to in a polynomial
/.leading coefficient - Answer-coefficient in front of the highest power of x
, /.# horizontal asymptotes for rational functions - Answer-1
/.graphs get interrupted by discontinuities - Answer-hole, single-point jump, piecewise
equations jumps, vertical asymptotes
/.removeable discontinuity - Answer-if you could redefine just one point, can turn into
function with no breaks
/.non-removeable discontinuity - Answer-limit 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) - Answer-if 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 - Answer-vector (direction and magnitude)
/.speed - Answer-scalar (only magnitude)
/.average speed - Answer-distance traveled/ change in time
/.average rate of change- "average" - Answer-a measure of what the "central" value is at
all our speed/rates at a given interval
secant line
/.instantaneous rate of change- "instantaneous" - Answer-a measure of what the value
is of your speed/rate at a given moment in time
tangent line
/.Strategy for drawing f'(x) from f(x) - Answer-1. 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 - Answer-limit 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 - Answer-numerical value for the slope of the tangent line
taking a continuous object and breaking it up to approximate (advantages,
disadvantages) - Answer-disadvantage- more pieces to deal with
advantage- more accurate estimate
/.Calculus - Answer-all about limits and the attempt of closing in on an exact value of an
answer by using finer and finer approximations
/.limit - Answer-o the 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 - Answer-look at what's happening to the y-coordinate as
you close in on a particular x-value
/.two-sided limits - Answer-function needs to be approaching the same y-value from
both the left and right to exist
/.one-sided limits - Answer-only focus on the y-value from one particular side
/.determine limits numerically - Answer-substitute 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) -
Answer-break single large limit into several smaller easier limits
/.Sandwich or Squeeze Theorem - Answer-if 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 - Answer-polynomial/polynomial
/.degree - Answer-highest power x is raised to in a polynomial
/.leading coefficient - Answer-coefficient in front of the highest power of x
, /.# horizontal asymptotes for rational functions - Answer-1
/.graphs get interrupted by discontinuities - Answer-hole, single-point jump, piecewise
equations jumps, vertical asymptotes
/.removeable discontinuity - Answer-if you could redefine just one point, can turn into
function with no breaks
/.non-removeable discontinuity - Answer-limit 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) - Answer-if 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 - Answer-vector (direction and magnitude)
/.speed - Answer-scalar (only magnitude)
/.average speed - Answer-distance traveled/ change in time
/.average rate of change- "average" - Answer-a measure of what the "central" value is at
all our speed/rates at a given interval
secant line
/.instantaneous rate of change- "instantaneous" - Answer-a measure of what the value
is of your speed/rate at a given moment in time
tangent line
/.Strategy for drawing f'(x) from f(x) - Answer-1. 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 - Answer-limit 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 - Answer-numerical value for the slope of the tangent line
