Week 7: Week In Review
MATH 131
3.1, 3.2, 3.3
DROST
-------------------------------------------------------------3.1 Derivative Rules 𝑓(𝑥) = 𝑐, 𝑓 ′ (𝑥) = 0
𝑓(𝑥) = 𝑚𝑥 + 𝑏, 𝑓 ′ (𝑥) = 𝑚
𝑓(𝑥) = 𝑎𝑥 𝑛 , 𝑓 ′ (𝑥) = (𝑎 ∗ 𝑛)𝑥 𝑛−1
𝑓(𝑥) = 𝑒 𝑥 , 𝑓 ′ (𝑥) = 𝑒 𝑥
Finding derivatives.
1. 𝑓(𝑥) = 3𝑥 2 + 𝑥 4 − 2√𝑥 + 9𝑒 𝑥 + 8
2. Write the equation of the tangent to the curve 𝑓(𝑥) = 3𝑥 2 − 𝑥 + 5 at 𝑥 = −1.
3. Write the equation of the normal to the curve 𝑓(𝑥) = 3𝑥 2 − 𝑥 + 5 at 𝑥 = −1.
4. State the x-values where the function 𝑓(𝑥) = 2𝑥 3 − 4𝑥 + 5, has a horizontal tangent line
Equations of motion: s = position, s’=velocity, s’’=acceleration
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5. Given 𝑠(𝑡) = 2𝑡 3 − 𝑡 + 9𝜋, where s is measured in feet, and t in seconds, find the acceleration
function at any time t, and at 2 seconds.
6. Find A, B, and C such that 𝑦 = 𝐴𝑥 2 + 𝐵𝑥 + 𝐶 satisfies the differential equation
𝑦 ′′ − 2𝑦 ′ + 𝑦 = 𝑥 2 + 𝑥 − 2.
3.2. Product and Quotient Rules
𝑦 = 𝑓 ∗ 𝑔,
𝑓
𝑦=𝑔 ,
𝑦 ′ = 𝑓 ∗ 𝑔′ + 𝑔 ∗ 𝑓′
𝑦′ =
𝑔∗𝑓 ′ −𝑓∗𝑔′
𝑔2
7. Find the derivative of 𝑓(𝑥) = (3𝑒 𝑥 + 1)(𝑥 2 − 2) .
8. Write the equation of the tangent at x=0, for 𝑓(𝑥) = (3𝑒 𝑥 + 1)(𝑥 2 − 2) .
𝑦 = 𝑒𝑥,
𝑦 ′ = 𝑒 𝑥 (ln 𝑒) = 𝑒 𝑥
𝑦 = 𝑏𝑥,
𝑦 ′ = 𝑏 𝑥 (ln 𝑏)
9. Find the derivative of 𝑦 = 𝑏𝑥 3 + 𝑏 2 + 𝑏 𝑥
10. Find the derivative where 𝑦 = (𝑥 2 − 𝑒𝑥 + 𝑏)(𝑥 − 𝑒 𝑥 + 3𝑥 ).
11. Find the derivative: 𝑦 =
4−3𝑥
𝑥+5
12. Given: 𝑓(𝑥) = √𝑥 ∗ 𝑔(𝑥), 𝑔(4) = 8, 𝑔′ (4) = −2 Find 𝑓′(4)
DNS
13. Given: 𝑓(𝑥) =
𝑥2
𝑔(𝑥)
, 𝑔(3) = −2, 𝑔′ (3) = 4. Find 𝑓′(3)
14. Find the equation of both lines that are tangent to 𝑦 = 𝑥 2 + 𝑥 that pass through (2, −3).
1+𝑒 𝑥
15. Find the equation of the tangent line to 𝑦 = 1+𝑥2 at 𝑥 = 0.
𝐹(𝑥)
16. Shown are the graphs of 𝐹(𝑥)𝑎𝑛𝑑 𝐺(𝑥). If 𝑃(𝑥) = 𝐹(𝑥) ∗ 𝐺(𝑥) 𝑎𝑛𝑑 𝑄(𝑥) = 𝐺(𝑥) ,
find 𝑃′ (2) 𝑎𝑛𝑑 𝑄′(7).
3.3 Derivatives of Trig Functions
𝑑
(sin 𝑥)
𝑑𝑥
= cos 𝑥
𝑑
(csc 𝑥)
𝑑𝑥
= −csc 𝑥 ∗ cot 𝑥
𝑑
(cos 𝑥)
𝑑𝑥
= − sin 𝑥
𝑑
(sec 𝑥)
𝑑𝑥
17. Find the derivative of: 𝑦 = 𝑥 3 ∗ cos 𝑥
18. Find the derivative of: 𝑦 = 𝑥 2 ∗ tan 𝑥
= sec 𝑥 ∗ tan 𝑥
𝑑
(tan 𝑥)
𝑑𝑥
= 𝑠𝑒𝑐 2 𝑥
𝑑
(cot 𝑥)
𝑑𝑥
= −𝑐𝑠𝑐 2 𝑥
1+sin 𝑥
19. Find the derivative of: 𝑦 = 1−𝑐𝑜𝑠 𝑥
20. Find the derivative of: 𝑦 =
cos 𝑥
𝑒𝑥
21. Find the derivative of: 𝑦 =
1+tan 𝑥
sec 𝑥
22. Show
𝑑
(sec 𝑥)
𝑑𝑥
= sec 𝑥 ∗ tan 𝑥
𝜋
23. Find the equation of the tangent to the curve 𝑦 = 3𝑥 + 6 cos 𝑥 at ( 3 , 𝜋 + 3)
24. Find the velocity and acceleration functions for 𝑠(𝑡) = 2 cos 𝑡 + 3 sin 𝑡 , where 𝑡 > 0,
s is measured in cm. and t in sec.
25. A ladder 10 feet long rests against a vertical wall. Let 𝜃 be the angle between the top of the
ladder and the wall, and let x be the distance from the bottom of the ladder to the wall. If the
bottom of the ladder slides away from the wall, how fast does x change with respect to 𝜃 when
𝜋
𝜃= ?
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