ISYE 6644 Master Calculus: Derivatives, Integrals, and Probability Concepts Questions and Answers | 2026 Update | 100% Correct.

Knowledge Check 2 Q If f ( x )=l n ( 2 x ) find the derivative f ' ( x ). 1/x Q If f ( x )=sin ( ln ( x ))ind the derivative f ' ( x ). cos ( ln ( x )) x Q Which of the following methods cannot be used to find the zeroes of a complicated function? Newman's method acting Trial-and-error: While inefficient, simply plugging in numbers to see if the output is zero (or close to it) is a valid, if primitive, method. Bisection: This is a reliable algorithm that repeatedly halves an interval and selects the subinterval in which a root must lie. It is slow but guaranteed to converge if the function is continuous and has opposite signs at the interval's endpoints. Newton's method: Also known as the Newton-Raphson method, this uses the function's derivative to quickly converge on a zero. It follows the tangent line of a point to the x axis to find the next approximation. Q Use your favorite numerical method to solve g ( x )=x2−3=0, for x ∈ [1,2] x = 1.732 x2−3=0 is equivalent to finding x=√3 ≈ 1.73205 Q Find 7/3 Q Find 1 ∫( x +1)2 dx. 0 2 ∫ln ( x ) dx. 1 2ln(2) - 1 x Q Find lim e −1 . (Hint: this problem will make you so sick, you’ll have to go to the …? x → 0 si n ( x ) 1 Q Which of the following is not an integration method discussed in this lesson? Newmann sums Q How does a mathematician capture a wild man-eating zoid? Select all that apply. You trap a zoid Trick question! It's always best to avoid a zoid altogether! 1 1 n 1 ∫ x2 dxusing the lesson’s form of the Riemann sum with n=4, specifically ∫ x2 dx ≈ ∑( ) i 2 Q Find the approximate value of the integral 15/32 0 0 n i=1 n Q Toss a 4-side die twice (you know, one of those goofy Dungeons and Dragons pyramid dice things). Assuming the die is numbered 1,2,3,4, what's the probability that the sum will equal 3? 18 Q f ( x )=3 e{−x } for x 0is a legitimate probability density function. False Non-negativity: f ( x ) ≥ 0 for all x. Normalization: The total area under the curve must equal exactly 1. Q Suppose X is a continuous random variable with cumulative distribution function F(x). What is the distribution of the nasty random variable F(X)? Unif (0,1) Q Suppose U is a Unif (0,1) random variable. Name the distribution of X =−l n (1−U ) Exponential Q TRUE or FALSE? 231−1 is a prime number. True Q Suppose X is a continuous random variable with p.d.f. f ( x )=3 x2 for 0 ≤ x ≤ 1. Find E [ X ].