AP Calculus BC Exam Review
QUESTIONS & ANSWERS(SCORED A+)
Jump Discontinuity - ANSWERWhen the two-sided limit doesn't exist because the
one-sided limits aren't equal.
Infinite Discontinuity - ANSWERWhen the two-sided limit doesn't exist because it's
unbounded.
Removable Discontinuity - ANSWERWhen the two-sided limit exists, but isn't equal
to the function's value.
Squeeze Theorem - ANSWERƒ(x) ≤ g(x) ≤ h(x) for all x and if the limit of ƒ(x) as x→c
= L and the limit of h(x) as x→c = L, then the limit of g(x) as x→c = L.
Intermediate Value Theorem - ANSWERIf f is continuous on [a,b] and k is a number
between f(a) and f(b), then there exists at least one number c such that f(c)=k.
Derivative - ANSWERThe derivative of a function describes the function's
instantaneous rate of change at a certain point.
Average Rate Of Change - ANSWERSlope of secant line between two points, used
to estimate instantaneous rate of change at a point.
Instantaneous Rate Of Change - ANSWERThe rate of change at a particular
moment.
Tangent Line - ANSWERA line that intersects a curve once and only once, slope of
tangent line is found by finding the derivative of the original line.
Secant Line - ANSWERA line that intersects a curve at two points. Basically the
average rate of change because it is the rate of change between two points on a
curve.
Power Rule - ANSWERnx^n-1
Product Rule - ANSWERf'(x)g(x)+f(x)g'(x)
Quotient Rule - ANSWERg(x)f'(x)-f(x)g'(x)/g(x)^2
d/dx sinx - ANSWERcosx
d/dx cosx - ANSWER-sinx
d/dx tanx - ANSWERsec^2x
d/dx secx - ANSWERsecxtanx
, d/dx cscx - ANSWER-cscxcotx
d/dx cotx - ANSWER-csc^2x
Implicit Differentiation - ANSWERThe process of finding the derivative of a
dependent variable in an implicit function by differentiating each term separately, by
expressing the derivative of the dependent variable as a symbol, and by solving the
resulting expression for the symbol.
Chain Rule - ANSWERd/dx f(g(x)) = f'(g(x)) g'(x)
Related Rates - ANSWERan equation involving two or more variables that are
differentiable functions of time can be used to find an equation that relates the
corresponding rates.
Velocity - ANSWERThe speed of an object in a particular direction, first derivative of
a position function.
Right Riemann Sum - ANSWERUses rectangles with right-endpoints to evaluate
integrals (estimate area). {Right} = Δx [ f (a + Δx) + f (a + 2 Δx) ... + f (b)]
Midpoint Riemann Sum - ANSWERFind the interval for x amount of rectangles. use
that midpoint times the change in x. n n ∑ i = 1f(x ∗ i)Δxi
Fundamental Theorem Of Calculus - ANSWERIf f is an integrable function and g(x) =
integral of f(x)dx, then the integral of f(x)dx from a to b = g(b) - g(a).
Acceleration - ANSWERThe rate at which velocity changes, second derivative of a
position function and first derivative or velocity function.
L'Hopital's Rule - ANSWERUsed to find indeterminate limits; find derivative of
numerator and denominator separately then evaluate limit.
Mean Value Theorem - ANSWERIf f(x) is continuous and differentiable, slope of
tangent line equals slope of secant line at least once in the interval (a, b)
f '(c) = [f(b) - f(a)]/(b - a)
Extreme Value Theorem - ANSWERIf f is continuous on [a,b] then f has an absolute
maximum and an absolute minimum on [a,b]. The global extrema occur at critical
points in the interval or at endpoints of the interval.
Critical Point - ANSWERPoint of the function at which the differential of the function
is zero or undefined. Can be calculated by setting the first derivative equal to zero.
Concavity - ANSWERDescribes the curvature of a function. A function is said to be
concave up if it curves upward, and concave down if it curves downward. The
concavity of a function can be determined by calculating its second derivative.
QUESTIONS & ANSWERS(SCORED A+)
Jump Discontinuity - ANSWERWhen the two-sided limit doesn't exist because the
one-sided limits aren't equal.
Infinite Discontinuity - ANSWERWhen the two-sided limit doesn't exist because it's
unbounded.
Removable Discontinuity - ANSWERWhen the two-sided limit exists, but isn't equal
to the function's value.
Squeeze Theorem - ANSWERƒ(x) ≤ g(x) ≤ h(x) for all x and if the limit of ƒ(x) as x→c
= L and the limit of h(x) as x→c = L, then the limit of g(x) as x→c = L.
Intermediate Value Theorem - ANSWERIf f is continuous on [a,b] and k is a number
between f(a) and f(b), then there exists at least one number c such that f(c)=k.
Derivative - ANSWERThe derivative of a function describes the function's
instantaneous rate of change at a certain point.
Average Rate Of Change - ANSWERSlope of secant line between two points, used
to estimate instantaneous rate of change at a point.
Instantaneous Rate Of Change - ANSWERThe rate of change at a particular
moment.
Tangent Line - ANSWERA line that intersects a curve once and only once, slope of
tangent line is found by finding the derivative of the original line.
Secant Line - ANSWERA line that intersects a curve at two points. Basically the
average rate of change because it is the rate of change between two points on a
curve.
Power Rule - ANSWERnx^n-1
Product Rule - ANSWERf'(x)g(x)+f(x)g'(x)
Quotient Rule - ANSWERg(x)f'(x)-f(x)g'(x)/g(x)^2
d/dx sinx - ANSWERcosx
d/dx cosx - ANSWER-sinx
d/dx tanx - ANSWERsec^2x
d/dx secx - ANSWERsecxtanx
, d/dx cscx - ANSWER-cscxcotx
d/dx cotx - ANSWER-csc^2x
Implicit Differentiation - ANSWERThe process of finding the derivative of a
dependent variable in an implicit function by differentiating each term separately, by
expressing the derivative of the dependent variable as a symbol, and by solving the
resulting expression for the symbol.
Chain Rule - ANSWERd/dx f(g(x)) = f'(g(x)) g'(x)
Related Rates - ANSWERan equation involving two or more variables that are
differentiable functions of time can be used to find an equation that relates the
corresponding rates.
Velocity - ANSWERThe speed of an object in a particular direction, first derivative of
a position function.
Right Riemann Sum - ANSWERUses rectangles with right-endpoints to evaluate
integrals (estimate area). {Right} = Δx [ f (a + Δx) + f (a + 2 Δx) ... + f (b)]
Midpoint Riemann Sum - ANSWERFind the interval for x amount of rectangles. use
that midpoint times the change in x. n n ∑ i = 1f(x ∗ i)Δxi
Fundamental Theorem Of Calculus - ANSWERIf f is an integrable function and g(x) =
integral of f(x)dx, then the integral of f(x)dx from a to b = g(b) - g(a).
Acceleration - ANSWERThe rate at which velocity changes, second derivative of a
position function and first derivative or velocity function.
L'Hopital's Rule - ANSWERUsed to find indeterminate limits; find derivative of
numerator and denominator separately then evaluate limit.
Mean Value Theorem - ANSWERIf f(x) is continuous and differentiable, slope of
tangent line equals slope of secant line at least once in the interval (a, b)
f '(c) = [f(b) - f(a)]/(b - a)
Extreme Value Theorem - ANSWERIf f is continuous on [a,b] then f has an absolute
maximum and an absolute minimum on [a,b]. The global extrema occur at critical
points in the interval or at endpoints of the interval.
Critical Point - ANSWERPoint of the function at which the differential of the function
is zero or undefined. Can be calculated by setting the first derivative equal to zero.
Concavity - ANSWERDescribes the curvature of a function. A function is said to be
concave up if it curves upward, and concave down if it curves downward. The
concavity of a function can be determined by calculating its second derivative.
