The Derivatives of the Trigonometric Functions Bernd Schr oder - PowerPoint PPT Presentation

Two Limits Derivatives Examples The Derivatives of the Trigonometric Functions Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ❑ ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ❑ θ ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ❑ θ ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ❑ θ ✲ x 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ r ❑ θ ✲ x 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) r ❑ θ ✲ x 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) r θ ❑ θ ✲ x 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) r θ ❑ θ ✲ x 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) r θ sin ( θ ) ❑ θ ✲ x 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) r θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) r θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r q θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r q P θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r q P θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ θ ≤ tan ( θ ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r q P θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ θ ≤ tan ( θ ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r q P θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ θ ≤ tan ( θ ) 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r q P θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ θ ≤ tan ( θ ) 1 ≥ sin ( θ ) θ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r q P θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ θ ≤ tan ( θ ) 1 ≥ sin ( θ ) = y θ θ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r q P θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ θ ≤ tan ( θ ) 1 ≥ sin ( θ ) = y y θ ≥ θ tan ( θ ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r q P θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ θ ≤ tan ( θ ) 1 ≥ sin ( θ ) = y y y tan ( θ ) = θ ≥ θ y / x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

Two Limits Derivatives Examples sin ( θ ) ≤ θ ≤ tan ( θ ) y ✻ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ( x , y ) tan ( θ ) r q P θ sin ( θ ) ❑ θ ✲ x 1 sin ( θ ) ≤ θ θ ≤ tan ( θ ) 1 ≥ sin ( θ ) = y y y tan ( θ ) = = x θ ≥ θ y / x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions