Derivatives Integrals Components of

Acceleration Equations

10

20

30

40 40

30

20

10

40

30

20

10

40

30

20

10

QUESTION:

𝑟 𝑡 = 2𝑡, 𝑡2 ANSWER:

𝑟 ′ 𝑡 = 2, 2𝑡

QUESTION:

𝑟 𝑡 = 𝑡𝑖 + sin 𝑡 𝑗 + cos 𝑡 𝑘 𝑎 𝜋 =?

ANSWER:

𝑎 𝑡 = 𝑘

QUESTION:

𝑟 𝑡 = 𝑒𝑡 , 𝑡2 − 𝑡, sin 𝑡

Find the tangent line at 𝑡 = 0.

ANSWER:

𝑥 = 1 + 𝑡; 𝑦 = −𝑡; 𝑧 = 𝑡

QUESTION:

Find 𝑑

𝑑𝑡(𝐹 𝑡 ∙ 𝐺 𝑡 ) at t=2 if:

𝐹 𝑡 = 3𝑡, 2𝑡2, 𝑒𝑡 , 𝐺 𝑡 = 0, ln 𝑥 ,1

3𝑡3

ANSWER:

4 +20

3𝑒2 + 8 ln 2

QUESTION:

𝑎 (𝑡) = 2𝑡, 𝑒𝑡 , 𝑣 0 = 0

ANSWER:

𝑣 𝑡 = 𝑡2, 𝑒𝑡 − 1

QUESTION:

𝑎 (𝑡) = 2𝑡, 𝑒𝑡 , 𝑣 0 =0

𝑟 (3) = 3, 𝑒3

ANSWER:

𝑟 (𝑡) =1

3𝑡3 − 6, 𝑒𝑡 − 𝑡 + 3

QUESTION:

𝑟 𝑡 = 𝑡, sin 𝑡 , cos 𝑡

Find the length of the curve from

0 ≤ 𝑡 ≤ 2.

ANSWER:

2 2

QUESTION:

𝑟 𝑡 = 2𝑡2,1

3𝑡3, 4𝑡

Find the length of the curve from

0 ≤ 𝑡 ≤ 2.

ANSWER:

32

3

QUESTION:

𝑟 (𝑡) = 3𝑡2 + 𝑡 − 2, 3𝑒2𝑡

𝑇 0 =? ANSWER:

1

37,

6

37

QUESTION:

𝑟 (𝑡) = 3𝑡2 + 𝑡 − 2, 3𝑒2𝑡 𝑎𝑇 0 =?

ANSWER:

78

37

QUESTION:

𝑟 (𝑡) = 3𝑡2 + 𝑡 − 2, 3𝑒2𝑡 𝑎𝑁 0 =?

ANSWER:

24

37

QUESTION:

𝑟 (𝑡) = 3𝑡2 + 𝑡 − 2, 3𝑒2𝑡

𝑁 0 =?

ANSWER:

6 37

37,−

37

37

QUESTION:

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦? ANSWER:

𝑑𝑟

𝑑𝑡 𝑜𝑟 𝑟 ′(𝑡)

QUESTION:

𝐶𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒?

ANSWER:

𝜅 =𝑇′(𝑡)

𝑟 ′(𝑡)

QUESTION:

𝑁𝑜𝑟𝑚𝑎𝑙 𝑉𝑒𝑐𝑡𝑜𝑟?

ANSWER:

𝑇′(𝑡)

𝑇′(𝑡) 𝑎𝑛𝑑

𝑎 − 𝑎𝑇𝑇

𝑎𝑁

QUESTION:

𝐶𝑖𝑟𝑐𝑙𝑒 𝑜𝑓 𝐶𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒?

ANSWER:

(𝑥 −𝑁𝑥

𝜅)2+(𝑦 −

𝑁𝑦

𝜅)2+(𝑧 −

𝑁𝑧

𝜅)2=

1

𝜅2