Calculus Midterm Review Exam Questions with Correct Grade A+ Solutions

Calculus: Midterm Review Exam Questions with Correct Grade A+ Solutions limits - Correct Answers: travel toward function from both sides and meet in the middle; right and left hand limits must be equal. can exist even with a hole in the graph. how do you find a limit? - Correct Answers: 1. substitution 2. factoring 3. conjugate method (radicals) a³ + b³ - Correct Answers: (a + b)(a² - ab + b²) a³ - b³ - Correct Answers: (a - b)(a² + ab + b²) vertical asymptote - Correct Answers: indicated by a nonzero number/0 horizontal asymptote - Correct Answers: if lm(x→∞) exists, f(x) = d has a horizontal asymptote at d limits involving infinity - Correct Answers: 1. if degree of numerator is equal to degree of denominator, lm(x→∞) = a/b 2. if degree of denominator is larger, lm(x→∞) = 0 3. if degree of numerator is larger, lm(x→∞) = ∞ (-∞ if difference is odd/negative, ∞ if difference is even) continuity - Correct Answers: no breaks, holes, jumps in graph limit at every x in domain no undefined points types of discontinuity - Correct Answers: point, infinite, jump intermediate value theorem - Correct Answers: if f(x) is continuous on [a,b], and if d is between them, then a corresponding c exists so f(c) = d lm(x→0) sin(x)/x - Correct Answers: 1 lm(x→0) 1-cos(x)/1 - Correct Answers: 0 lm(x→0) tan(x)/x - Correct Answers: 1 formal derivative definition - Correct Answers: lm(h→0) = f(a+h) - f(a) / h power rule - Correct Answers: y = axⁿ y' = (n)axⁿ⁻¹ derivative of a constant - Correct Answers: 0 product rule - Correct Answers: y = f(x) × g(x) y' = f(x) × g'(x) + g(x) × f'(x) quotient rule - Correct Answers: y = f(x)/g(x) y' = g(x) × f'(x) - f(x) × g'(x) / [g(x)]² chain rule - Correct Answers: [f(g(x))]' = f'(g(x)) × g'(x) (derivative of the outside)(derivative of the inside) sin(x) - Correct Answers: cos(x) cos(x) - Correct Answers: -sin(x) tan(x) - Correct Answers: sec²(x) csc(x) - Correct Answers: -csc(x)cot(x) sec(x) - Correct Answers: sec(x)tan(x) cot(x) - Correct Answers: -csc²(x) e^x - Correct Answers: (e^x)dx first derivative test - Correct Answers: 1. find zeros if possible 2. find critical numbers (zeros of the first derivative) 3. make interval table (stepping around critical numbers) to determine intervals increasing/decreasing 4. graph potential spikes - Correct Answers: zeros of denominators (if sign changes) second derivative test - Correct Answers: zeros of second derivative are potential inflection points; make an interval table stepping around critical numbers to find changes in concavity. volume of a cone - Correct Answers: V = 1/3πr²h volume of a sphere - Correct Answers: V = 4/3πr³ volume of a cylinder - Correct Answers: V = πr²h surface area of a cylinder - Correct Answers: SA = 2πrh + 2πr² logbm + logbn - Correct Answers: logb(mn) logbm - logbn - Correct Answers: logb(m/n) logbm^p - Correct Answers: plogbm change of base - Correct Answers: logbm = logm/logb d/dx(a^x) - Correct Answers: (a^x)ln(a)dx graphing derivatives - Correct Answers: max/min → derivative has a zero increasing/decreasing → derivative is above/below x-axis related rates - Correct Answers: 1. find an equation that relates the varying quantities. 2. differentiate both sides of the equation with respect to time, obtaining an equation that relates the various rates of change. 3. substitute in values for all known quantities and derivatives. 4. solve for the remaining rate. optimization - Correct Answers: 1. draw an appropriate figure and label the quantities relevant to the problem. 2. find a formula for the quantity to be maximized or minimized. 3. using the conditions stated in the problem to eliminate variables, express the quantity to be maximized or minimized as a function of one variable. 4. find the interval of possible values for this variable from the physical restrictions in the problem. linearization - Correct Answers: L(x) = f(a) + f'(a)(x-a) graph of velocity - Correct Answers: velocity is increasing/decreasing when above/below axis acceleration is the slope find the points on the curve where the lines tangent to the curve are vertical - Correct Answers: undefined slope -> set denominator of slope equal to 0