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AP Calculus AB: 5.1.4 The Number Pi

Mathematics16 CardsCreated 8 months ago

This flashcard set explores the mathematical constant π (pi), its geometric definition, and its appearance in trigonometric functions. It covers how π is used in derivatives involving trigonometric expressions through the Chain Rule and includes applications in geometry and identifying irrational numbers.

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The Number Pi

  • 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.

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Key Terms

Term
Definition

The Number Pi

  • 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 term...

note

  • 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 ...

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

(3π+6) ft

Find the derivative of the given function.

f(x)=cscπx^2

f′(x)=−2πxcotπx^2cscπx^2

Find the derivative of the given function.

f(x)=tan2πx

f′(x)=2πsec^2*2πx

Which of the following numbers is not irrational?

3.3¯

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AP Calculus AB: 5.1.4 The Number Pi
TermDefinition

The Number Pi

  • 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.

note

  • 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.

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

(3π+6) ft

Find the derivative of the given function.

f(x)=cscπx^2

f′(x)=−2πxcotπx^2cscπx^2

Find the derivative of the given function.

f(x)=tan2πx

f′(x)=2πsec^2*2πx

Which of the following numbers is not irrational?

3.3¯

Find the derivative of the given function.

f(x)=tan(π√x)

f′(x)=π√xsec^2 (π√x) / 2x

Find the derivative of the given function.

f(x)=secπx

f′(x)=πtanπxsecπx

Find the derivative of the given function.

f(x)=sin(π/x)

f′(x)=−πcos(π/x) / x^2

Find the derivative of the given function.

f(x)=cosπx^2

f′(x)=−2πxsinπx^2

Find the derivative of the given function.

f(x)=3sin2πx

f′(x)=6πcos2πx

Find the derivative of the given function.

f(x)=xsinπx / π

f′(x)=sinπx+πxcosπx / π

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

9π/2 ft2

Find the derivative of the given function.

f(x)=sin(cos2πx)

f′(x)=−2πsin2πxcos(cos2πx)

Find the derivative of the given function.

f(x)=4πsin2πxcos2πx

f′(x)=8π^2(cos^22πx−sin^22πx)

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

A=C^2/4π