AP Calculus AB Exam Questions and Answers

AP Calculus AB Exam Intermediate Value Theorem - Answer-If f(1)=-4 and f(6)=9, then there must be a x-value between 1 and 6 where f crosses the x-axis. Average Rate of Change - Answer-Slope of secant line between two points, use to estimate instantanous rate of change at a point. Instantenous Rate of Change - Answer-Slope of tangent line at a point, value of derivative at a point Formal definition of derivative - Answer-limit as h approaches 0 of [f(a+h)-f(a)]/h 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 fro 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 Product Rule - Answer-uv' + vu' Quotient Rule - Answer-(uv'-vu')/v² 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) - 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 absolute value of velocity - Answer-speed y = sin(x), y' = - Answer-y' = cos(x) y = cos(x), y' = - Answer-y' = -sin(x) y = tan(x), y' = - Answer-y' = sec²(x) y = csc(x), y' = - Answer-y' = -csc(x)cot(x) y = sec(x), y' = - Answer-y' = sec(x)tan(x) y = cot(x), y' = - Answer-y' = -csc²(x) y = sin⁻¹(x), y' = - Answer-y' = 1/√(1 - x²) y = cos⁻¹(x), y' = - Answer-y' = -1/√(1 - x²) y = tan⁻¹(x), y' = - Answer-y' = 1/(1 + x²) y = cot⁻¹(x), y' = - Answer-y' = -1/(1 + x²) y = e^x, y' = - Answer-y' = e^x y = a^x, y' = - Answer-y' = a^x ln(a) y = ln(x), y' = - Answer-y' = 1/x y = log (base a) x, y' = - Answer-y' = 1/(x lna) To find absolute maximum on closed interval [a, b], you must consider... - Answer-critical points and endpoints 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) 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 Linearization - Answer-use tangent line to approximate values of the function rate - Answer-derivative left riemann sum - Answer-use rectangles with left-endpoints to evaluate integral (estimate area) right riemann sum