Calculus Answers

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

Question 9

In a bottle, the height of the liquid is a function of the volume of the liquid in the bottle, say h = 𝑔(𝑣). Similarly, the volume of the liquid is function of the height of the liquid, say 𝑣 = 𝑓(h). Show that 𝑓and 𝑔 are inverse functions.

Question 10

Scientists have developed a radioisotope-based power supply whose power output in watts is

125 given by the following function: 𝑃 = 75𝑒− 𝑡

where t is the number of days the power supply is used. The experimental power supply has been installed in an unmanned undersea vehicle on an exploration mission. The vehicle requires 7watts to operate at full capacity. Use the given information to find the number of days for which the vehicle can operate at full capacity.

Question 6

Classify each of the following functions as algebraic or transcendental.

a) 𝑓(𝑥) = tan(8𝑥)

b) 𝑓(𝑥) = √𝑥5+3 9𝑥+5

c) 𝑓(𝑥) = 9𝑥

d) 𝑓(𝑥) = 1+𝑥

A company that operates for 18 hours a day has introduced a new daily wage system that incorporates variable hourly rate to encourage longer shifts. The company pays $10 for first hour or any part of the first hour. Afterwards, company pays additional 2% per hour for each hour or part of each additional hour. Write a piece-wise function that shows the salary as a function of hours. Also sketch a graph of this function.

Question 8

Describe how the function 𝑓(𝑥) = −(𝑥 + 1)2 − 4 can be graphed using the graph of 𝑦 = 𝑥2 and a sequence of transformations.