Calculus Answers

Consider the following equation "I=\\displaystyle{\\int_{0}^{1}\\sin{(x^2)}dx}"

Find the integral  "\\displaystyle{\\int_{-1}^{1}\\sin{(x^2)}dx}" in terms of "I"

True or false:

The binomial coefficient "\\displaystyle{{-1\/2}\\choose{5}}" is equal to "\\dfrac{63}{256}"

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Consider the following three functions:

1. "x^2"
2. "xcosx"
3. "e^x"

Which of them is a) An even function, b) AN odd function and c) neither an even or odd function?

True or false:

If the function  f is  odd, then "\\displaystyle{\\int_{-1}^{1}f(x)dx=0}"

Evaluate the limit "\\displaystyle{\\lim\\limits_{\\theta \\to 0} \\dfrac{\\sin{(\\theta^2)}}{\\theta}}" using the l'Hopital's Rule.

Evaluate the limit "\\displaystyle{\\lim\\limits_{x \\to 0} \\dfrac{e^{x}-x-1}{x^2}}" by using l'Hopital's Rule twice.

1. Find a formula for the function whose graph consists of the line segment from the point (−4,3) to the point (−2,0) and the lower half of the circle centered at the origin with radius 2.

Use chain rule to find the derivative and express the final answer in terms of x in radical form of

y= u^1/2 and u= x^1/2

1.Find the second and third derivatives in the simplest form of

y=x⁵-3x²-2x+5

2. Find by implicit differentiation of

z²-3zy+y²=6z-2y

Question 19

Identify the domain of the following functions.

a) 𝑓(𝑥) = sin−1(3𝑥 − 1)

b) 𝑓(𝑥) = [log(sin−1(√𝑥2 + 3𝑥 + 2))]

c)𝑘(𝑥)= 1 √(𝑥−2)2

d) 𝑗(𝑥)= 1 𝑥−√(𝑥+2)

e) 𝑓(𝑥)=ln|𝑥+3|−5