Math 151 WIR 5: Sections 3.2, 3.3, 3.4, 3.5, 3.6. a. b.

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© Dr. Rosanna Pearlstein, Spring 2015
Math 151 WIR 5: Sections 3.2, 3.3, 3.4, 3.5, 3.6.
1. Find the derivative of each of the following functions.
a. 𝑦 = π‘₯ 5 βˆ’ πœ‹ 3 βˆ’ 4π‘₯ 2 + 2
5
3
b. 𝑦 = √π‘₯ βˆ’ √π‘₯ 2 βˆ’ 5
√π‘₯
c. 𝑦 = (3π‘₯ 5 + π‘₯ 4 βˆ’ πœ‹ 2 + 1)(π‘₯ βˆ’1 βˆ’ π‘₯ βˆ’2 + π‘₯ βˆ’3 )
1
1
π‘₯
π‘₯2
π‘₯
5
d. 𝑦 = βˆ’
+
1
π‘₯3
1
e. 𝑦 = 5 + π‘₯ + 5π‘₯ + 5π‘₯
2. Find an equation of the tangent line to the curve 𝑦 =
3. Find the derivative of the function 𝐻(π‘₯) =
4. Find the point(s) on the graph of 𝑦 =
π‘₯
π‘₯βˆ’3
(𝑓(π‘₯)+1)3
𝑓(π‘₯)βˆ’2π‘₯
π‘₯βˆ’1
π‘₯ 2 +1
at π‘₯ = 2.
, where f is a differentiable function. Do not simplify.
where the tangent line is perpendicular to the line 𝑦 = 3π‘₯ βˆ’ 1.
5. A particle moves in a straight line so that its position x cm from 0 at time t seconds is given by
π‘₯(𝑑) = 𝑑 3 βˆ’ 9𝑑 2 + 24𝑑 + 2.
a. At what times is the particle at rest?
b. When is the particle moving in the positive direction?
c. Find the total distance traveled from π‘₯ = 0 to π‘₯ = 6
d. Find the displacement from π‘₯ = 0 to π‘₯ = 6.
6. Evaluate the following limits:
πœƒ
a. limπœƒβ†’0 𝑠𝑖𝑛3πœƒ
𝑠𝑖𝑛5πœƒ
b. limπœƒβ†’0 𝑠𝑖𝑛3πœƒ
c. limπ‘₯β†’0
𝑠𝑖𝑛2 5π‘₯
3π‘₯ 2
d. lim𝑑→0
e. limπ‘₯β†’0
f. limπ‘₯β†’3
(π‘π‘œπ‘ π‘‘)βˆ’1
tan(2𝑑)
sin(π‘π‘œπ‘ π‘₯)
π‘π‘œπ‘ π‘₯
sin(3βˆ’π‘₯)
(π‘₯ 2 βˆ’9)
© Dr. Rosanna Pearlstein, Spring 2015
7. Find the derivative of each function:
π‘₯βˆ’π‘ π‘–π‘›π‘₯
a. 𝑓(π‘₯) = 𝑠𝑒𝑐π‘₯+π‘‘π‘Žπ‘›π‘₯
b. 𝑔(π‘₯) = 𝑐𝑠𝑐π‘₯π‘π‘œπ‘‘π‘₯ βˆ’ π‘₯π‘π‘œπ‘ π‘₯
c. β„Ž(π‘₯) = tan(𝑓(π‘₯)), where f is a differentiable function.
d. π‘˜(π‘₯) = 𝑠𝑖𝑛(√π‘₯ 2 + π‘₯) βˆ’ 𝑠𝑒𝑐 2 π‘₯ + βˆšπ‘π‘œπ‘ (𝑠𝑖𝑛π‘₯)
e. 𝑙(π‘₯) = 𝑠𝑖𝑛2 π‘₯ + π‘π‘œπ‘  2 π‘₯
f. 𝑠(π‘₯) = 𝑠𝑒𝑐 2 π‘₯ βˆ’ π‘‘π‘Žπ‘›2 π‘₯
g. π‘Ÿ(π‘₯) =
1
π‘₯+
1
1
π‘₯+
π‘₯
8. Assume that y is a function of x. Find y' = dy/dx for (π‘₯ βˆ’ 𝑦)2 = π‘₯ + 𝑦 βˆ’ 1.
9. Assume that y is a function of x. Find y' = dy/dx for 𝑦 = sin(3π‘₯ + 4𝑦).
2
3
3
2
10. Assume that y is a function of x. Find y' = dy/dx for y = x y + x y .
11. Assume that y is a function of x. Find y' = dy/dx for
12. Find points on the curve 𝑦 =
π‘π‘œπ‘ π‘₯
2+𝑠𝑖𝑛π‘₯
𝑦
π‘₯3
π‘₯
+ 𝑦3 = π‘₯ 2 𝑦 4.
0 ≀ π‘₯ ≀ 2Ο€ where the tangent line is horizontal.
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