AP Calculus BC Exam Review Questions
and Answers
Average Rate of Change - Correct Answers -Slope of secant line between two points,
use to estimate instantanous rate of change at a point.
Instantenous Rate of Change - Correct Answers -Slope of tangent line at a point, value
of derivative at a point
Definition of Derivative - Correct Answers -limit as h approaches 0 of [f(a+h)-f(a)]/h or
limit as x approaches a of [f(x)-f(a)]/(x-a)
When f '(x) is positive, f(x) is - Correct Answers -increasing
When f '(x) is negative, f(x) is - Correct Answers -decreasing
When f '(x) changes from negative to positive, f(x) has a - Correct Answers -relative
minimum
When f '(x) changes fro positive to negative, f(x) has a - Correct Answers -relative
maximum
When f '(x) is increasing, f(x) is - Correct Answers -concave up
When f '(x) is decreasing, f(x) is - Correct Answers -concave down
When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has
a - Correct Answers -point of inflection
When is a function not differentiable - Correct Answers -corner, cusp, vertical tangent,
discontinuity
Product Rule - Correct Answers -uv' + vu'
Quotient Rule - Correct Answers -(uv'-vu')/v²
Chain Rule - Correct Answers -f '(g(x)) g'(x)
Particle is moving to the right/up - Correct Answers -velocity is positive
Particle is moving to the left/down - Correct Answers -velocity is negative
,absolute value of velocity - Correct Answers -speed
y = sin(x), y' = - Correct Answers -y' = cos(x)
y = cos(x), y' = - Correct Answers -y' = -sin(x)
y = tan(x), y' = - Correct Answers -y' = sec²(x)
y = csc(x), y' = - Correct Answers -y' = -csc(x)cot(x)
y = sec(x), y' = - Correct Answers -y' = sec(x)tan(x)
y = cot(x), y' = - Correct Answers -y' = -csc²(x)
y = sin⁻¹(x), y' = - Correct Answers -y' = 1/√(1 - x²)
y = cos⁻¹(x), y' = - Correct Answers -y' = -1/√(1 - x²)
y = tan⁻¹(x), y' = - Correct Answers -y' = 1/(1 + x²)
y = cot⁻¹(x), y' = - Correct Answers -y' = -1/(1 + x²)
y = e^x, y' = - Correct Answers -y' = e^x
y = a^x, y' = - Correct Answers -y' = a^x ln(a)
y = ln(x), y' = - Correct Answers -y' = 1/x
y = log (base a) x, y' = - Correct Answers -y' = 1/(x lna)
To find absolute maximum on closed interval [a, b], you must consider... - Correct
Answers -critical points and endpoints
Linearization - Correct Answers -use tangent line to approximate values of the function
left riemann sum - Correct Answers -use rectangles with left-endpoints to evaluate
integral (estimate area)
right riemann sum - Correct Answers -use rectangles with right-endpoints to evaluate
integrals (estimate area)
Trapezoidal rRle - Correct Answers -use trapezoids to evaluate integrals (estimate area)
[(h1 - h2)/2]*base - Correct Answers -area of trapezoid
, definite integral - Correct Answers -has limits a & b, find antiderivative, F(b) - F(a)
indefinite integral - Correct Answers -no limits, find antiderivative + C, use inital value to
find C
area under a curve - Correct Answers -∫ f(x) dx integrate over interval a to b
area above x-axis is - Correct Answers -positive
area below x-axis is - Correct Answers -negative
average value of f(x) - Correct Answers -= 1/(b-a) ∫ f(x) dx on interval a to b
To find particular solution to differential equation, dy/dx = x/y - Correct Answers -
separate variables, integrate + C, use initial condition to find C, solve for y
To draw a slope field, - Correct Answers -plug (x,y) coordinates into differential
equation, draw short segments representing slope at each point
slope of horizontal line - Correct Answers -zero
slope of vertical line - Correct Answers -undefined
methods of integration - Correct Answers -substitution, parts, partial fractions
∫ u dv = - Correct Answers -uv - ∫ v du
dP/dt = kP(M - P) - Correct Answers -logistic differential equation, M = carrying capacity
P = M / (1 + Ae^(-Mkt)) - Correct Answers -logistic growth equation
given rate equation, R(t) and inital condition when
t = a, R(t) = y₁ find final value when t = b - Correct Answers -y₁ + Δy = y
Δy = ∫ R(t) over interval a to b
given v(t) and initial position t = a, find final position when t = b - Correct Answers -s₁+
Δs = s
Δs = ∫ v(t) over interval a to b
given v(t) find displacement - Correct Answers -∫ v(t) over interval a to b
given v(t) find total distance travelled - Correct Answers -∫ abs[v(t)] over interval a to b
area between two curves - Correct Answers -∫ f(x) - g(x) over interval a to b, where f(x)
is top function and g(x) is bottom function
and Answers
Average Rate of Change - Correct Answers -Slope of secant line between two points,
use to estimate instantanous rate of change at a point.
Instantenous Rate of Change - Correct Answers -Slope of tangent line at a point, value
of derivative at a point
Definition of Derivative - Correct Answers -limit as h approaches 0 of [f(a+h)-f(a)]/h or
limit as x approaches a of [f(x)-f(a)]/(x-a)
When f '(x) is positive, f(x) is - Correct Answers -increasing
When f '(x) is negative, f(x) is - Correct Answers -decreasing
When f '(x) changes from negative to positive, f(x) has a - Correct Answers -relative
minimum
When f '(x) changes fro positive to negative, f(x) has a - Correct Answers -relative
maximum
When f '(x) is increasing, f(x) is - Correct Answers -concave up
When f '(x) is decreasing, f(x) is - Correct Answers -concave down
When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has
a - Correct Answers -point of inflection
When is a function not differentiable - Correct Answers -corner, cusp, vertical tangent,
discontinuity
Product Rule - Correct Answers -uv' + vu'
Quotient Rule - Correct Answers -(uv'-vu')/v²
Chain Rule - Correct Answers -f '(g(x)) g'(x)
Particle is moving to the right/up - Correct Answers -velocity is positive
Particle is moving to the left/down - Correct Answers -velocity is negative
,absolute value of velocity - Correct Answers -speed
y = sin(x), y' = - Correct Answers -y' = cos(x)
y = cos(x), y' = - Correct Answers -y' = -sin(x)
y = tan(x), y' = - Correct Answers -y' = sec²(x)
y = csc(x), y' = - Correct Answers -y' = -csc(x)cot(x)
y = sec(x), y' = - Correct Answers -y' = sec(x)tan(x)
y = cot(x), y' = - Correct Answers -y' = -csc²(x)
y = sin⁻¹(x), y' = - Correct Answers -y' = 1/√(1 - x²)
y = cos⁻¹(x), y' = - Correct Answers -y' = -1/√(1 - x²)
y = tan⁻¹(x), y' = - Correct Answers -y' = 1/(1 + x²)
y = cot⁻¹(x), y' = - Correct Answers -y' = -1/(1 + x²)
y = e^x, y' = - Correct Answers -y' = e^x
y = a^x, y' = - Correct Answers -y' = a^x ln(a)
y = ln(x), y' = - Correct Answers -y' = 1/x
y = log (base a) x, y' = - Correct Answers -y' = 1/(x lna)
To find absolute maximum on closed interval [a, b], you must consider... - Correct
Answers -critical points and endpoints
Linearization - Correct Answers -use tangent line to approximate values of the function
left riemann sum - Correct Answers -use rectangles with left-endpoints to evaluate
integral (estimate area)
right riemann sum - Correct Answers -use rectangles with right-endpoints to evaluate
integrals (estimate area)
Trapezoidal rRle - Correct Answers -use trapezoids to evaluate integrals (estimate area)
[(h1 - h2)/2]*base - Correct Answers -area of trapezoid
, definite integral - Correct Answers -has limits a & b, find antiderivative, F(b) - F(a)
indefinite integral - Correct Answers -no limits, find antiderivative + C, use inital value to
find C
area under a curve - Correct Answers -∫ f(x) dx integrate over interval a to b
area above x-axis is - Correct Answers -positive
area below x-axis is - Correct Answers -negative
average value of f(x) - Correct Answers -= 1/(b-a) ∫ f(x) dx on interval a to b
To find particular solution to differential equation, dy/dx = x/y - Correct Answers -
separate variables, integrate + C, use initial condition to find C, solve for y
To draw a slope field, - Correct Answers -plug (x,y) coordinates into differential
equation, draw short segments representing slope at each point
slope of horizontal line - Correct Answers -zero
slope of vertical line - Correct Answers -undefined
methods of integration - Correct Answers -substitution, parts, partial fractions
∫ u dv = - Correct Answers -uv - ∫ v du
dP/dt = kP(M - P) - Correct Answers -logistic differential equation, M = carrying capacity
P = M / (1 + Ae^(-Mkt)) - Correct Answers -logistic growth equation
given rate equation, R(t) and inital condition when
t = a, R(t) = y₁ find final value when t = b - Correct Answers -y₁ + Δy = y
Δy = ∫ R(t) over interval a to b
given v(t) and initial position t = a, find final position when t = b - Correct Answers -s₁+
Δs = s
Δs = ∫ v(t) over interval a to b
given v(t) find displacement - Correct Answers -∫ v(t) over interval a to b
given v(t) find total distance travelled - Correct Answers -∫ abs[v(t)] over interval a to b
area between two curves - Correct Answers -∫ f(x) - g(x) over interval a to b, where f(x)
is top function and g(x) is bottom function
