5.1.4 The Number Pi

Question 1

The Number Pi

Answer 1

• Pi (π) can be defined as the circumference of a circle whose diameter is 1. π is an irrational number, so its decimal expression never terminates or repeats.
• The area of a circle can be approximated by a rectangle.
• Trigonometric functions often involve the number pi. Differentiating such functions requires the use of the Chain Rule.

Question 2

note

Answer 2

• Consider a circle with a diameter of length 1. If you measure the circumference of the circle, you will find that it is approximately 3.14 times as long as the diameter. This ratio holds for any circle you draw. Mathematicians have a special name for this constant number. They call it pi (π).
• π is an irrational number. An irrational number is one that cannot be expressed as a fraction. As a result, the decimal expression of π does not terminate or repeat. The numbers go on forever without pattern.
• You can derive the area of a circle by cutting it up and
  rearranging the pieces. Half of the circumference makes up the width and the radius makes up the height. The more partitions you break the circle into the more the circle resembles a rectangle. If you cut the circle into an infinite number of slices, the circle would fit inside the rectangle exactly.
• The number pi arises in trigonometry because there are 2π radians in a circle. An example of a common trigonometric function is the function f(x) shown here.
• To find the derivative of this function, apply the Chain Rule. The derivative of sin(blop) is cos(blop), so the derivative of f(x) is cos(blop) times the derivative of the blop.
• Remember, pi is just a number, so it should be treated like any other number. In this case, it is part of the coefficient that is multiplied by x.

Question 3

A man built a platform in the shape of a semicircle with a radius of 3 feet. What is the perimeter of the platform?

Answer 3

(3π+6) ft

Question 4

Find the derivative of the given function.

f(x)=cscπx^2

Answer 4

f′(x)=−2πxcotπx^2cscπx^2

Question 5

Find the derivative of the given function.

f(x)=tan2πx

Answer 5

f′(x)=2πsec^2*2πx

Question 6

Which of the following numbers is not irrational?

Answer 6

3.3¯

Question 7

Find the derivative of the given function.

f(x)=tan(π√x)

Answer 7

f′(x)=π√xsec^2 (π√x) / 2x

Question 8

Find the derivative of the given function.

f(x)=secπx

Answer 8

f′(x)=πtanπxsecπx

Question 9

Find the derivative of the given function.

f(x)=sin(π/x)

Answer 9

f′(x)=−πcos(π/x) / x^2

Question 10

Find the derivative of the given function.

f(x)=cosπx^2

Answer 10

f′(x)=−2πxsinπx^2

Question 11

Find the derivative of the given function.

f(x)=3sin2πx

Answer 11

f′(x)=6πcos2πx

Question 12

Find the derivative of the given function.

f(x)=xsinπx / π

Answer 12

f′(x)=sinπx+πxcosπx / π

Question 13

A man built a platform in the shape of a semicircle with a radius of 3 feet. What is the area of the platform?

Answer 13

9π/2 ft2

Question 14

Find the derivative of the given function.

f(x)=sin(cos2πx)

Answer 14

f′(x)=−2πsin2πxcos(cos2πx)

Question 15

Find the derivative of the given function.

f(x)=4πsin2πxcos2πx

Answer 15

f′(x)=8π^2(cos^22πx−sin^22πx)

Question 16

Which of the following equations correctly relates the circumference and the area of a circle?

Answer 16

A=C^2/4π