Calculus Exam Questions with Correct Grade A+ Solutions

Calculus Exam Questions with Correct Grade A+ Solutions State the 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. State the Chain Rule for Differentiation - Correct Answers: n(f(x))^n-1 * f'(x) What is the integral of ln(x)? - Correct Answers: ... What is the Product Rule? - Correct Answers: [f(x) * g'(x)] + [f('x) * g(x)] Symbolically write the Quotient Rule. - Correct Answers: [(f'(x) * g(x)) - ((f(x) * g'(x))] / (g(x))^2 How do you find critical points? - Correct Answers: Set f' equal to 0 What does an inflection point signify? - Correct Answers: A change in concavity Does the second derivative need to exist for there to be an inflection point? - Correct Answers: No, the second derivative could be undefined How do you find a local maximum? - Correct Answers: Set the derivative and set it equal to zero (or undefined) and check it. How do you find a local minimum? - Correct Answers: Set the derivative and set it equal to zero (or undefined) and check it. Take the derivative of sin(x). - Correct Answers: cos(x) Take the derivative of cos(x). - Correct Answers: -sin(x) Take the derivative of ln(x). - Correct Answers: 1/x Take the derivative of csc(x). - Correct Answers: -csc(x)cot(x) Take the derivative of arccsc(x). - Correct Answers: -u' / |u| √(u^2 - 1) Take the derivative of arccot(x). - Correct Answers: -u' / 1 + u^2 Take the derivative of cot(x). - Correct Answers: -csc^2(x) Take the derivative of sec(x). - Correct Answers: sec(x)tan(x) Take the derivative of arcsec(x). - Correct Answers: u' / |u| √(u^2 - 1) Take the derivative of a^x. - Correct Answers: ln(a) * a^x Take the derivative of log∨a(x). - Correct Answers: 1 / ln(a) * x Take the derivative of arctan(x). - Correct Answers: u' / 1+u^2 Take the derivative of tan(x). - Correct Answers: sec^2(x) Take the derivative of arccos(x). - Correct Answers: -u' / √(1 - u^2) Take the derivative of e^x. - Correct Answers: e^x Take the derivative of arcsin(x). - Correct Answers: u' / √1-u^2) What is sin(0)? - Correct Answers: 0 What is sin 30⁰? - Correct Answers: (1/2) What is tan(π)? - Correct Answers: 0 What is 30 degrees in radians? - Correct Answers: 0.524 R State the Fundamental Theorem of Calculus. - Correct Answers: ∫ from a - b of f(x) dx = F(b) - F(a) where F'(x) = f(x) What is the Mean Value Theorem? - Correct Answers: If the function f(x) is continuous on [a,b], and it is also differentiable on the open set (a,b), you can always find an x = c in (a,b) such that f'(c) = [f(b) - f(a)] / (b-a) How do you find the average velocity? - Correct Answers: final position - initial position / total time