# 3 11 sin 3 1 3cos 3 1 12 sin 7 5 5 cos 7 5 13 cos 3 3

• 136

This preview shows page 24 - 30 out of 136 pages.

3 ?
11. ? = sin( 3? + 1) ? = 3cos (3? + 1) 12. ? = sin( 7 − 5? ) ? = −5 cos (7 − 5?) 13. ? = cos( 3 ?) ? = − 3 sin ( 3 ?) 14. ? = tan (2? − ? 3 ) ? = (2 − 3? 2 )??? 2 (2? − ? 3 ) 15. ? = cos( ????) ? = −???? sin (????) Question 6: Find the equation of the tangent to the curve at the indicated point ? = 2???? , ? = 2???? , ? = ? 4
Calculus Page 25 of 136 Subtopic 2.5 Derivatives of Trigonometric Functions Trigonometric Derivatives ( ???? ) = ???? ( ???? ) = ?′???? ( ???? ) = −???? ( ???? ) = −?′???? ( ???? ) = ??? 2 ? ( ???? ) = ?′ ??? 2 ? ( ???? ) = ??? 2 ? ( ???? ) = −?′ ??? 2 ? ( ???? ) = ???? tan ? ( ???? ) = ? ???????? ( ???? ) = −???? ???? ( ???? ) = −?′???? ???? Question 1: Find ?? ?? 1. ?(?) = 2? − 3???? ? ( ? ) = 2 − 3???? 2. ?(?) = 3???? − ? ? ( ? ) = 3??? 2 ? − 1 3. ?(?) = 10 ???? − 2???? ? ( ? ) = 10??? 2 ? + 2??? 2 ? 4. ?(?) = 1 2?𝑖??−4 cos ? ? ( ? ) = ( 0 )( 2???? − 4???? ) − (2 ???? + 4????)(1) (2???? − 4????) 2 ? ( ? ) = −2????+4???? (2????−4????) 2 5. ?(?) = tan ? ???? ? ( ? ) = ( ??? 2 ? )( ???? ) + (???????? )(???? ) ? ( ? ) = ??? 3 ? + ???? ??? 2 ? ? ( ? ) = ???? (??? 2 ? + ??? 2 ?) 6. ?(?) = ???? 1+?𝑖?? ? ( ? ) = ( −???? )( 1 + ???? ) − (????)(????) (1 + ????) 2 ? ( ? ) = −???? ??? 2 ? ??? 2 ? (1+????) 2 ??? 2 𝜃 + ??? 2 𝜃 = 1 ? ( ? ) = −???? − 1 (1 + ????) 2 7. ?(?) = 3? + ????? ? ( ? ) = 3 + [( 1 )( ???? ) + (??? 2 ?)(?) ] ? ( ? ) = 3 + ???? + ???? 2 ? Question 2:
Find an equation of the tangent line to the curve ? = 4 + ??? ? − 2???? at 𝑃( ? 2 , 2) ? = −??? 2 ? + 2???? ???? ? = −??? 2 ? 2 + 2??? ? 2 ??? ? 2 ? = −1 ?𝑖? 2 𝜋 2 + 2 ?𝑖? 𝜋 2 ??? 𝜋 2 ?𝑖? 𝜋 2 = −1 1 + 2 1 0 1 = −1 ? = ?(? − ? 1 ) + ? 1 ? = −1 (? − ? 2 ) + 2 ? = −? + ? 2 + 2 Question 3: A spring hanging from the ceiling vibrates up and down. Its vertical position at time t (in sec) is given by ?(?) = 3???2? ( in cm) a. Find the velocity of the spring at time t. .
b. What is the spring's maximum speed?
c. For what time values is the speed at its maximum ?
d. What is its location when it reaches its maximum speed ?
Calculus Page 26 of 136 Question 4:
A spring hanging from the ceiling vibrates up and down. Its vertical position at time t (in sec) is given by ?(?) = 4???3? ( in cm) a. Find the velocity of the spring at time t. .
b. What is the spring's maximum speed?
c. For what time values is the speed at its maximum ?
d. What is its location when it reaches its maximum speed ?
Question 5: A body is moving in simple harmonic motion with position function ? = ???? − ???? Find the jerk at time ?
Calculus Page 27 of 136 Subtopic 2.6 Derivatives of Exponential and Logarithmic Functions
Calculus Page 28 of 136 Exponential and logarithmic Derivatives ( ? ? ) = ? ? ( ? ? ) = ?′ ? ? ( ? ? ) = ? ? ??? ( ? ? ) = ?′ ? ? ??? ( ???) = 1 ? ( ???) = ?′ ? ( log ? ?) = 1 ???? ( log ? ?) = ?′ ???? Question 1 : Find ?? ?? 1. ?(?) = ( 1 4 ) ? 2 ( ? ? ) = ?′ ? ? ??? ? = ? 2 → ? = 2? ?′(?) = (2?) ( 1 4 ) ? 2 ln ( 1 4 ) = 2? ln ( 1 4 ) ( 1 4 ) ? 2 2. ?(?) = (5) ? 2 ( ? ? ) = ?′ ? ? ??? ? = ? 2 → ? = 2? ?′(?) = (2?)(5) ? 2 (ln 5) = 2? ln 5 (5) ? 2 3. ?(?) = log 10 (2? − 3) ( log ? ?) = ?′ ???? ? = 2? − 3 → ? = 2 ?′(?) = 2 (2?−3)ln 10 4. ?(?) = ?. ? 3 − 2? ? ?′(?) = ? 3 − 2? ? 5. ?(?) = 2 ???? ( ? ? ) = ?′ ? ? ??? ? = cot ? → ? = −??? 2 ? ?′(?) = (−??? 2 ?)(2 ???? )(??2) = −??? 2 ? ln 2 (2 ???? ) 6. ?(?) = log 4 ? 2 ( log ? ?) = ?′ ???? ? = ? 2 → ? = 2? ?′(?) = 2? ? 2 ln 4 = 2 ??? 4 7. ?(?) = ln (? 2 )
Calculus Page 29 of 136 ( ???) = ?′ ? ? = ? 2 → ? = 2? ?′(?) = 2? ? 2 = 2 ? 8. ?(?) = ( ???) 2 ?′(?) = (2)(???) ( 1 ? ) = 2 ln? ? 9. ?(?) = ln (???) ( ???) = ?′ ? ? = ln ? → ? = 1 ? ?′(?) = ( 1 ? ) ??? = 1 ??? ? 10. ?(?) = ln (??? ? 2 ) ( ???) = ?′ ? ? = ???? 2 → ? = −2????? 2 ?′(?) = −2??𝑖?? 2 ???? 2 = −2? ???? 2 11. ?(?) = sin (?? ? 2 ) ? = ??? 2 → ? = 2? ? 2 = 2 ? ?′(?) = 2 ? cos (??? 2 ) Question 2 : Find all the values of ? for which the tangent line to ? = 2? ?