AP Calculus BC Exam Review And
Answers
Formal definition of derivative - Answer-
/.Alternate definition of derivative - Answer-limit as x approaches a of [f(x)-f(a)]/(x-a)
/.When f '(x) is positive, f(x) is - Answer-increasing
/.When f '(x) is negative, f(x) is - Answer-decreasing
/.When f '(x) changes from negative to positive, f(x) has a - Answer-relative minimum
/.When f '(x) changes from positive to negative, f(x) has a - Answer-relative maximum
/.When f"(x) is increasing, f(x) is - Answer-concave up
/.When f"(x) is decreasing, f(x) is - Answer-concave down
/.When f"(x) changes from increasing to decreasing or decreasing to increasing, f(x) has
a - Answer-point of inflection
/.When is a function not differentiable - Answer-corner, cusp, vertical tangent,
discontinuity
/.Quotient Rule - Answer-(uv'-vu')/v² ........... or LO dHI - HI dLO / (LO)(LO)
/.Chain Rule - Answer-f '(g(x)) g'(x)
/.y = x cos(x), state rule used to find derivative - Answer-product rule
/.y = ln(x)/x², state rule used to find derivative - Answer-quotient rule
/.y = cos²(3x), state rule used to find derivative - Answer-chain rule
/.Particle is moving to the right/up - Answer-velocity is positive
/.Particle is moving to the left/down - Answer-velocity is negative
/.To find absolute maximum on closed interval [a, b], you must consider... - Answer-
critical points and endpoints
, /.If f '(x) = 0 and f"(x) > 0, - Answer-f(x) has a relative minimum
/.If f '(x) = 0 and f"(x) < 0, - Answer-f(x) has a relative maximum
/.Slope of the tangent line at a given point (a, b) - Answer-(y - b) = y'(a) (x - a)
/.a rate is the same as a - Answer-derivative
/.left riemann sum - Answer-use rectangles with left-endpoints to evaluate integral
(estimate area)
/.right riemann sum - Answer-use rectangles with right-endpoints to evaluate integrals
(estimate area)
/.Local Linear Approximation - Answer-f(a) + f'(a)(x-a)
/.definite integral - Answer-has limits a & b, to find antiderivative, F(b) - F(a)
/.indefinite integral - Answer-no limits, find antiderivative + C, use inital value (given
(x,y)) to find C
/.area under a curve - Answer-∫ f(x) dx integrate over interval a to b
/.area above x-axis is - Answer-positive
/.area below x-axis is - Answer-negative
/.average value of f(x) - Answer-= 1/(b-a) ∫ f(x) dx on interval a to b
/.If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) = - Answer-g'(x) = f(x)
/.Fundamental Theorem of Calculus - Answer-∫ f(x) dx on interval a to b = F(b) - F(a)
/.To find particular solution to differential equation, dy/dx = x/y - Answer-separate
variables, integrate + C, use initial condition to find C, solve for y
/.To draw a slope field, - Answer-plug (x,y) coordinates into differential equation, draw
short segments representing slope at each point
/.slope of horizontal line - Answer-0
/.slope of vertical line - Answer-undefined
/.methods of integration - Answer-u-sub, parts, trig-sub, partial fractions
Answers
Formal definition of derivative - Answer-
/.Alternate definition of derivative - Answer-limit as x approaches a of [f(x)-f(a)]/(x-a)
/.When f '(x) is positive, f(x) is - Answer-increasing
/.When f '(x) is negative, f(x) is - Answer-decreasing
/.When f '(x) changes from negative to positive, f(x) has a - Answer-relative minimum
/.When f '(x) changes from positive to negative, f(x) has a - Answer-relative maximum
/.When f"(x) is increasing, f(x) is - Answer-concave up
/.When f"(x) is decreasing, f(x) is - Answer-concave down
/.When f"(x) changes from increasing to decreasing or decreasing to increasing, f(x) has
a - Answer-point of inflection
/.When is a function not differentiable - Answer-corner, cusp, vertical tangent,
discontinuity
/.Quotient Rule - Answer-(uv'-vu')/v² ........... or LO dHI - HI dLO / (LO)(LO)
/.Chain Rule - Answer-f '(g(x)) g'(x)
/.y = x cos(x), state rule used to find derivative - Answer-product rule
/.y = ln(x)/x², state rule used to find derivative - Answer-quotient rule
/.y = cos²(3x), state rule used to find derivative - Answer-chain rule
/.Particle is moving to the right/up - Answer-velocity is positive
/.Particle is moving to the left/down - Answer-velocity is negative
/.To find absolute maximum on closed interval [a, b], you must consider... - Answer-
critical points and endpoints
, /.If f '(x) = 0 and f"(x) > 0, - Answer-f(x) has a relative minimum
/.If f '(x) = 0 and f"(x) < 0, - Answer-f(x) has a relative maximum
/.Slope of the tangent line at a given point (a, b) - Answer-(y - b) = y'(a) (x - a)
/.a rate is the same as a - Answer-derivative
/.left riemann sum - Answer-use rectangles with left-endpoints to evaluate integral
(estimate area)
/.right riemann sum - Answer-use rectangles with right-endpoints to evaluate integrals
(estimate area)
/.Local Linear Approximation - Answer-f(a) + f'(a)(x-a)
/.definite integral - Answer-has limits a & b, to find antiderivative, F(b) - F(a)
/.indefinite integral - Answer-no limits, find antiderivative + C, use inital value (given
(x,y)) to find C
/.area under a curve - Answer-∫ f(x) dx integrate over interval a to b
/.area above x-axis is - Answer-positive
/.area below x-axis is - Answer-negative
/.average value of f(x) - Answer-= 1/(b-a) ∫ f(x) dx on interval a to b
/.If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) = - Answer-g'(x) = f(x)
/.Fundamental Theorem of Calculus - Answer-∫ f(x) dx on interval a to b = F(b) - F(a)
/.To find particular solution to differential equation, dy/dx = x/y - Answer-separate
variables, integrate + C, use initial condition to find C, solve for y
/.To draw a slope field, - Answer-plug (x,y) coordinates into differential equation, draw
short segments representing slope at each point
/.slope of horizontal line - Answer-0
/.slope of vertical line - Answer-undefined
/.methods of integration - Answer-u-sub, parts, trig-sub, partial fractions
