AP Calculus BC Exam Review Questions
With Verified Answers
Jump Discontinuity - Answer-When the two-sided limit doesn't exist because the one-
sided limits aren't equal.
/.Infinite Discontinuity - Answer-When the two-sided limit doesn't exist because it's
unbounded.
/.Removable Discontinuity - Answer-When the two-sided limit exists, but isn't equal to
the function's value.
/.Squeeze Theorem - Answer-ƒ(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 - Answer-If 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 - Answer-The derivative of a function describes the function's instantaneous
rate of change at a certain point.
/.Average Rate Of Change - Answer-Slope of secant line between two points, used to
estimate instantaneous rate of change at a point.
/.Instantaneous Rate Of Change - Answer-The rate of change at a particular moment.
/.Tangent Line - Answer-A 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 - Answer-A 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 - Answer-nx^n-1
/.Product Rule - Answer-f'(x)g(x)+f(x)g'(x)
/.Quotient Rule - Answer-g(x)f'(x)-f(x)g'(x)/g(x)^2
/.d/dx sinx - Answer-cosx
/.d/dx cosx - Answer--sinx
, /.d/dx tanx - Answer-sec^2x
/.d/dx secx - Answer-secxtanx
/.d/dx cscx - Answer--cscxcotx
/.d/dx cotx - Answer--csc^2x
/.Implicit Differentiation - Answer-The 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 - Answer-d/dx f(g(x)) = f'(g(x)) g'(x)
/.Related Rates - Answer-an 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 - Answer-The speed of an object in a particular direction, first derivative of a
position function.
/.Acceleration - Answer-The rate at which velocity changes, second derivative of a
position function and first derivative or velocity function.
/.L'Hopital's Rule - Answer-Used to find indeterminate limits; find derivative of numerator
and denominator separately then evaluate limit.
/.Mean Value Theorem - Answer-If 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 - Answer-If 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 - Answer-Point 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 - Answer-Describes 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.
/.d/dx sin-1x - Answer-1/√ 1-x^2
With Verified Answers
Jump Discontinuity - Answer-When the two-sided limit doesn't exist because the one-
sided limits aren't equal.
/.Infinite Discontinuity - Answer-When the two-sided limit doesn't exist because it's
unbounded.
/.Removable Discontinuity - Answer-When the two-sided limit exists, but isn't equal to
the function's value.
/.Squeeze Theorem - Answer-ƒ(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 - Answer-If 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 - Answer-The derivative of a function describes the function's instantaneous
rate of change at a certain point.
/.Average Rate Of Change - Answer-Slope of secant line between two points, used to
estimate instantaneous rate of change at a point.
/.Instantaneous Rate Of Change - Answer-The rate of change at a particular moment.
/.Tangent Line - Answer-A 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 - Answer-A 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 - Answer-nx^n-1
/.Product Rule - Answer-f'(x)g(x)+f(x)g'(x)
/.Quotient Rule - Answer-g(x)f'(x)-f(x)g'(x)/g(x)^2
/.d/dx sinx - Answer-cosx
/.d/dx cosx - Answer--sinx
, /.d/dx tanx - Answer-sec^2x
/.d/dx secx - Answer-secxtanx
/.d/dx cscx - Answer--cscxcotx
/.d/dx cotx - Answer--csc^2x
/.Implicit Differentiation - Answer-The 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 - Answer-d/dx f(g(x)) = f'(g(x)) g'(x)
/.Related Rates - Answer-an 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 - Answer-The speed of an object in a particular direction, first derivative of a
position function.
/.Acceleration - Answer-The rate at which velocity changes, second derivative of a
position function and first derivative or velocity function.
/.L'Hopital's Rule - Answer-Used to find indeterminate limits; find derivative of numerator
and denominator separately then evaluate limit.
/.Mean Value Theorem - Answer-If 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 - Answer-If 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 - Answer-Point 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 - Answer-Describes 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.
/.d/dx sin-1x - Answer-1/√ 1-x^2
