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Math 11-Worksheet 07 - Worksheet

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Calculus I (MATH 011)

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University of California, Merced

Academic year: 2022/2023

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Math 11 Worksheet 7 for Sections 3, 3, and 3 Spring 2023

1 Section 3.

All I see are patterns: Take the derivative of each of the following functions. You might need the Chain Rule at first, but try to find an underlying pattern.

y = sin θ, y = sin 2θ, y = sin 3θ, y = sin 4θ, y = sin πθ

All I see are patterns (again): Take the derivative of each of the following functions. You might need the Chain Rule at first, but try to find an underlying pattern.

y =

√

x, y =

√

2 + sin x, y =

√

3 + cos x, y =

√

4 + tan x, y =

√

5 + 6x 7

Chain of fools: Take the derivative of the following functions. You may need to utilize the Chain Rule. It might be helpful to first decompose each expression first.

(a) A(x) = (4x − x 2 ) 100

(b) B(x) = (1 + x 4 ) 2 / 3

√ 3

1 + tan t

(d) D(θ) = a 3 + cos 3 θ

(e) E(θ) = 4 cos(nθ)

(f) F (p) = (3p − 1) 4 (2p + 1)− 3

(g) G(s) =

√

s 2 + 1

s 2 + 4

(h) H(x) =

x √ 7 − 3 x

(i) I(θ) = sin

(√

1 + 10θ

)

(j) J(x) = cos 2 (sin x)

Decomposition: (i) Find functions f , g, and h such that F = f ◦ g ◦ h = f (g(h(x))). Note: there might be multiple answers (come up with at least one answer for each F ). (ii) Find the derivative, F ′(x), of each.

(a) F (x) =

√

1 − √x

(b) F (x) = sin 3 (2x + 3)

1

(2x 2 + x + 3) 3

(d) F (x) =

√

x + 1 − 1 √ x + 1 + 1

Off on a tangent Find the equation of the tangent line to the curve at the given point.

(a) f (x) =

√

1 + x 3 , (2, 3)

(b) f (θ) = sin θ + sin 2 θ, (0, 0)

(d) f (θ) = tan 2 θ, (π/ 4 , 1)

(e) f (x) = csc x, (π/ 6 , 2)

(f) f (θ) = cot θ, (π/ 6 ,

√

2)

2 Section 3.

Find the given quantity with the information provided.

(a)

dx f

− 1 (2), given f − 1 (2) = 0 and f ′ (0) = − 2

(b)

dx f

− 1 (6), given f ′(−1) = 6 and f − 1 (6) = − 1

dx f

− 1 (0) = 2/3 and f − 1 (0) = 3

1

Math 11 Worksheet 7 for Sections 3, 3, and 3 Spring 2023

Given the information and indicated function, find

dx f

− 1 (a) for the given value of a.

(a) f (x) = x 3 + x + 1 at a = 1. Note: f (0) = 1

(b) f (x) = 1 − x 2 − x 3 at a = 5. Note: f (−2) = 5.

x + 1

x − 2

at a = −3. Note: f (1) = −3.

(d) f (x) =

− 3 x + 2

− 3 − x

at a = 14. Note: f (−4) = 14.

(e) f (x) =

√

− 1 − 5 x at a = 7. Note: f (−10) = 7.

(f) f (x) =

√

3 x + 4 at a = 2. Note: f (0) = 2.

Find the derivative of the given functions.

(a) A(x) = sin − 1 x − cos − 1 x

(b) B(x) = (x 2 + 1) tan − 1 x

(d) D(x) =

x + 1

arccos(2x + 1)

(e) E(x) = sin(arccos x)

(f) F (x) =

√

arctan(7 − 3 x)

Find the equation of the tangent line to the given function at x = a.

(a) f (x) = x tan − 1 x, a = 1

(b) g(x) = x 2 − arccos(1 + x), a = − 1

− 1 (x 2 )

x + 1

, a = 0

3 Section 3.

Troubleshooting: What is wrong with the following calculations?

f (x) = 6 x

⇒ f ′ (x) = (ln x) x

⇒ f

′ (2) = (ln 2)

= 36 ln 2

= ln 2 36

Troubleshooting What is wrong with the following calculations? What should the derivative be?

y = 7

x 3

⇒ y ′ = 7

x (3x 2 ) − (ln 7) x x 3

x 6

A probe leaving a planet has a position function given by r(t) = R 0 + e t −

1

2

mgt 2 , where R 0 is the

radius of the planet, m is the mass of the probe, g is a constant related to the gravitational pull of the planet, and t is time, in seconds. Find the (a) velocity and (b) accleration functions for the probe.

2

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Math 11-Worksheet 07 - Worksheet

Course: Calculus I (MATH 011)

81 Documents

Students shared 81 documents in this course

University: University of California, Merced

Was this document helpful?

Math 11 Worksheet 7 for Sections 3.6, 3.7, and 3.9 Spring 2023

1 Section 3.6

1. All I see are patterns: Take the derivative of each of the following functions. You might need the

Chain Rule at first, but try to find an underlying pattern.

y= sin θ, y = sin 2θ, y = sin 3θ, y = sin 4θ, y = sin πθ

2. All I see are patterns (again): Take the derivative of each of the following functions. You might

need the Chain Rule at first, but try to find an underlying pattern.

y=√x, y =√2 + sin x, y =√3 + cos x, y =√4 + tan x, y =p5 + 6x7

3. Chain of fools: Take the derivative of the following functions. You may need to utilize the Chain

Rule. It might be helpful to first decompose each expression first.

(a) A(x) = (4x−x2)100

(b) B(x) = (1 + x4)2/3

√1 + tan t

(d) D(θ) = a3+ cos3θ

(e) E(θ) = 4 cos(nθ)

(f) F(p) = (3p−1)4(2p+ 1)−3

(g) G(s) = rs2+ 1

s2+ 4

(h) H(x) = x

√7−3x

(i) I(θ) = sin √1 + 10θ

(j) J(x) = cos2(sin x)

4. Decomposition: (i) Find functions f,g, and hsuch that F=f◦g◦h=f(g(h(x))). Note: there

might be multiple answers (come up with at least one answer for each F). (ii) Find the derivative,

F′(x), of each.

(a) F(x) = p1−√x

(b) F(x) = sin3(2x+ 3)

(2x2+x+ 3)3

(d) F(x) = √x+ 1 −1

√x+ 1 + 1

5. Off on a tangent Find the equation of the tangent line to the curve at the given point.

(a) f(x) = √1 + x3,(2,3)

(b) f(θ) = sin θ+ sin2θ, (0,0)

(d) f(θ) = tan2θ, (π/4,1)

(e) f(x) = csc x, (π/6,2)

(f) f(θ) = cot θ, (π/6,√2)

2 Section 3.7

1. Find the given quantity with the information provided.

(a) d

dxf−1(2), given f−1(2) = 0 and f′(0) = −2

(b) d

dxf−1(6), given f′(−1) = 6 and f−1(6) = −1

dxf−1(0) = 2/3 and f−1(0) = 3

Math 11-Worksheet 07 - Worksheet

Calculus I (MATH 011)

University of California, Merced

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1 Section 3.

√

√

√

√

√

√ 3

√

(√

)

√

1

√

√

√

2)

2 Section 3.

1

√

√

√

3 Section 3.

1

2

2

Math 11-Worksheet 07 - Worksheet

Course: Calculus I (MATH 011)

University: University of California, Merced

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