Calculus Memorize Test A
Derivatives:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
(𝑥𝑛 ) =
(𝑓(𝑥)𝑔(𝑥)) =
𝑓( 𝑥 )
( )=
𝑑𝑥 𝑔(𝑥)
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
(𝑓(𝑔(𝑥))) =
Integrals:
1. ∫ 𝑥𝑛 𝑑𝑥 =
2. ∫ sin 𝑥 𝑑𝑥 =
3. ∫ cos 𝑥 𝑑𝑥 =
4. ∫ sec2 𝑥 𝑑𝑥 =
5. ∫ sec 𝑥 tan 𝑥 𝑑𝑥 =
(sin 𝑥) =
6. ∫ csc 𝑥 cot 𝑥 𝑑𝑥 =
(cos 𝑥) =
7. ∫ csc2 𝑥 𝑑𝑥 =
(tan 𝑥) =
8. ∫ 𝑒𝑥 𝑑𝑥 =
9. ∫ 𝑎𝑥 𝑑𝑥 =
(sec 𝑥) =
1
(csc 𝑥) =
10. ∫ 𝑥 𝑑𝑥 =
(cot 𝑥) =
(𝑒𝑥 ) =
(ln 𝑥) =
Alternative Definition of Derivative:
𝑓 ′ (𝑐) =
Instantaneous Rate of Change IRoC (Formal definition of
derivative with “h”):
Average Rate of Change of 𝑓(𝑥) on the interval [𝑎, 𝑏]:
The Fundamental Theorem of Calculus
𝑏
∫ 𝑓(𝑥)𝑑𝑥 =
𝑎
Corrollary to FTC
𝑔(𝑥)
𝑑
∫ 𝑓(𝑡)𝑑𝑡 =
𝑑𝑥
𝑎
Where 𝐹 ′ (𝑥) =
Intermediate Value Theorem:
Mean Value Theorem:
Area of a Trapezoid
Average Value of 𝑓 on [𝑎, 𝑏]:
The relationship between position, velocity, and
acceleration:
Displacement of 𝑓 on [𝑎, 𝑏]:
Total Distance Travelled:
Difference between Speed and Velocity
ln(𝑒) =
Volumes of Rotation (Disc & Washer):
ln(1) =
𝜋
sin ( ) =
2
𝜋
cos ( ) =
4
𝜋
sin ( ) =
6
Calculus Memorize Test B
Derivatives:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
𝑑
𝑑𝑥
(𝑥𝑛 ) =
𝑑
(𝑓(𝑥)𝑔(𝑥)) =
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑓( 𝑥 )
(𝑔(𝑥)) =
Integrals:
11. ∫ 𝑥𝑛 𝑑𝑥 =
12. ∫ 𝑒𝑥 𝑑𝑥 =
13. ∫ 𝑎𝑥 𝑑𝑥 =
1
14. ∫ 𝑑𝑥 =
𝑥
(𝑓(𝑔(𝑥))) =
15. ∫ sin 𝑥 𝑑𝑥 =
(𝑒𝑥 ) =
16. ∫ cos 𝑥 𝑑𝑥 =
(ln 𝑥) =
17. ∫ sec2 𝑥 𝑑𝑥 =
(sin 𝑥) =
18. ∫ sec 𝑥 tan 𝑥 𝑑𝑥 =
(cos 𝑥) =
19. ∫ csc 𝑥 cot 𝑥 𝑑𝑥 =
20. ∫ csc2 𝑥 𝑑𝑥 =
(tan 𝑥) =
(sec 𝑥) =
(csc 𝑥) =
Area of a Trapezoid
(cot 𝑥) =
Average Value of 𝑓 on [𝑎, 𝑏]:
Average Rate of Change of 𝑓(𝑥) on the interval [𝑎, 𝑏]:
The relationship between position, velocity, and
acceleration:
Displacement of 𝑓 on [𝑎, 𝑏]:
Alternative Definition of Derivative:
Instantaneous Rate of Change IRoC (Formal definition of
derivative with “h”):
𝑓 ′ (𝑐) =
The relationship between position, velocity, and
acceleration:
Displacement of 𝑓 on [𝑎, 𝑏]:
The Fundamental Theorem of Calculus
Corrollary to FTC
𝑏
∫ 𝑓(𝑥)𝑑𝑥 =
𝑎
𝑔(𝑥)
𝑑
∫ 𝑓(𝑡)𝑑𝑡 =
𝑑𝑥
𝑎
Where 𝐹 ′ (𝑥) =
Total Distance Travelled:
Difference between Speed and Velocity
ln(𝑒) =
Volumes of Rotation (Disc & Washer):
ln(1) =
sin (
3𝜋
)=
4
𝜋
cos ( ) =
2
𝜋
sin ( ) =
6
Intermediate Value Theorem:
Mean Value Theorem: