AP Calculus Review Exam Questions with Correct Grade A+ Solutions

AP Calculus Review Exam Questions with Correct Grade A+ Solutions Critical Point - Correct Answers: f'(x) = 0 or undefined (and endpoints on closed interval) local minimum - Correct Answers: f'(x) goes from - to + local maximum - Correct Answers: f'(x) goes from + to - point of inflection - Correct Answers: - concavity changes - f''(x) goes from - to + or + to - d/dx(xⁿ) - Correct Answers: nxⁿ⁻¹ d/dx(sinx) - Correct Answers: cosx d/dx(cosx) - Correct Answers: -sinx d/dx(tanx) - Correct Answers: sec²x d/dx(cotx) - Correct Answers: -csc²x d/dx(secx) - Correct Answers: secxtanx d/dx(cscx) - Correct Answers: -cscxcotx d/dx(lnn) - Correct Answers: (1/n)(dn/dx) [w/ n acting as "u"] d/dx(eⁿ) - Correct Answers: (eⁿ)(dn/dx) [w/ n acting as "u"] d/dx [arcsin(u/a)] - Correct Answers: (1/√a²-u²)(du/dx) d/dx[arccosx] - Correct Answers: -1/√1-x² d/dx[arctan(u/a)] - Correct Answers: (a/a²+u²)(du/dx) d/dx[arccotx] - Correct Answers: -1/1 + x² d/dx[arcsec(u/a)] - Correct Answers: (a/|u|√u²-a²)(du/dx) d/dx[arccscx] - Correct Answers: -1/|x|√x²-1 d/dx(aⁿ) - Correct Answers: (aⁿ)(lna)(dn/dx) [w/ n acting as "u"] d/dx(log₀x) - Correct Answers: 1/xln0 [w/ 0 acting as "a"] Chain rule: d/dx[f(u)] - Correct Answers: f'(u)(du/dx) Product rule: d/dx(uv) - Correct Answers: u'v + uv' Quotient rule: d/dx(u/v) - Correct Answers: (u'v - uv')/v² Fundamental Theorem of Calculus - Correct Answers: ∫f(x)dx from a to b = F(a) - F(b) [where F'(x) = f(x)] Intermediate Value Theorem - Correct Answers: If the function f(x) is continuous on [a,b], and y is a number between f(a) and f(b), then there exists at least one number x = c in the open interval (a,b) such that f(c) = y Mean Value Theorem - Correct Answers: If the function f(x) is continuous on [a,b], and the first derivative exists on the interval (a,b), then there exists at least one number x = c in the open interval (a,b) such that f'(c) = [f(b) - f(a)]/b-a Rolle's Theorem - Correct Answers: If the function f(x) is continuous n [a,b], and the first derivative exists on the interval (a), and f(a) = f(b), then there is at least 1 number x = c in (a,b) such that f'(c) = 0 Average Value of a Function - Correct Answers: f(c) = (1/b-a)∫f(x)dx on a to b Disk Method - Correct Answers: V = π∫[R(x)]² on x=a to x=b Washer Method - Correct Answers: V = π∫([R(x)]²-[r(x)]²)dx General Volume equation (not rotated) - Correct Answers: V = ∫Area(x)dx on a to b velocity - Correct Answers: d/dt(position) acceleration - Correct Answers: d/dt(velocity) displacement - Correct Answers: ∫vdt from 0 to t distance - Correct Answers: ∫|v|dt from initial time to final time average velocity - Correct Answers: ∆x/∆t L'Hopital's Rule - Correct Answers: If f(x)/g(b) = 0/0 or ±∞/±∞, then the limit as x approaches a of f(x)/g(x) = the limit as x approaches a of f'(x)/g'(x) sin2x - Correct Answers: 2sinxcosx cos2x - Correct Answers: cos²x - sin²x or 1-2sin²x cos²x - Correct Answers: 1/2(1 + cos2x) sin²x - Correct Answers: 1/2(1-cos2x) sin²x + cos²x - Correct Answers: 1 1 + tan²x - Correct Answers: sec²x cot²x + 1 - Correct Answers: csc²x ∫tanxdx - Correct Answers: ln|secx| + C or -ln|cosx| + C ∫secxdx - Correct Answers: ln|secx + tanx| + C