ISYE 6644 Master Calculus: Derivatives, Integrals, and Probability
Concepts
Knowledge Check 2
Q If f ( x )=l n ( 2 x ) find the derivative f ' ( x ).
1/x
Q Iff ( 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
1
Q Find ∫ ( x +1)2 dx.
0
7/3
2
Q Find ∫ ln ( x ) dx.
1
2ln(2) - 1
e x −1
Q Find lim . (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! n 2
1 1 i
∑( )
1
Q Find the approximate value of the integral ∫x 2
dxusing the lesson’s form of the Riemann sum with n=4, specifically ∫x 2
dx ≈
0 0 n i=1 n
15/32
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?
1\8
Concepts
Knowledge Check 2
Q If f ( x )=l n ( 2 x ) find the derivative f ' ( x ).
1/x
Q Iff ( 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
1
Q Find ∫ ( x +1)2 dx.
0
7/3
2
Q Find ∫ ln ( x ) dx.
1
2ln(2) - 1
e x −1
Q Find lim . (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! n 2
1 1 i
∑( )
1
Q Find the approximate value of the integral ∫x 2
dxusing the lesson’s form of the Riemann sum with n=4, specifically ∫x 2
dx ≈
0 0 n i=1 n
15/32
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?
1\8
