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Math 11-Worksheet 10 Sol

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Calculus I (MATH 011)

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University of California, Merced

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1 Section 4.

For the given functions, where is it concave up/down? Give your answers in interval notation. You may use the work done from Worksheet 9! (a) f (x) = x 3 + 3x 2 − 1

❼ f ′′(x) = 6x + 6 and setting f ′′(x) = 0, we get x = −1. f (x) is concave up on [− 1 , ∞) and concave down on (−∞, −1]. (b) g(x) =

√

3 x − 2 sin x, on (0, 2 π)

❼ g′′(x) = 2 sin x and setting g′′(x) = 0, we get x = π (x = 0 is not in the interval). g(x) is concave up on (0, π] and concave down on [π, 2 π).

x 2 + 3 x + 1

❼ h′′(x) =

8

(x + 1) 3

and setting h′′(x) = 0, we find that there are no solutions, but still a critical value at x = −1. h(x) is concave up on (− 1 , ∞) and concave down on (−∞, −1).

Gaussians: (part 2) In Math 15, Math 18, and/or Psychology 10; you will encounter a Gaussian curve—colloquially known as a “bell curve”. The standard, normal distribution is defined by the function f (z) =

1 √

2 π e

− 0. 5 z 2

Familiarize yourself with the shape 1 of the function by analyzing the function for properties such as the following: (a) Over what intervals is the function concave up? ... concave down? Are there any points of inflection? You may use the work done from Worksheet 9! Here, f ′′(z) =

z 2 − 1 √ 2 π e

−z 2 / 2.

CU on (−∞, −1]∪[1, ∞) and CD on [− 1 , 1]. Points of inflection are (− 1 , 1 /

√

2 eπ) and (1, 1 /

√

2 eπ) 3. Sigmoid Function: (part 2) In a branch of computer science called Machine Learning, scientists like to employ sigmoid functions to transform data. 2 Analyze the function

f (z) =

1

1 + e−z for properties such as the following: (a) Over what intervals is the function concave up? ... concave down? Are there any points of inflection? You may use the work done from Worksheet 9! Here, f ′′(z) = −

e−z (1 − e−z ) (1 + e−z ) 3

.

CU on (−∞, 0] and CD on [0, ∞). Point of inflection is (0, 1 /2). 1 Further (optional) reading: The reason for the coefficient √ 1 2 π is to ensure that the area between the curve and the horizontal axis is equal to one. Considering that the bell curve is used in the study of probabilities, why is it important for that area to be equal to one? 2 Further (optional) reading: The goal of a sigmoid function is to map all possible inputs to a relatively small range of numbers. Furthermore, a sigmoid function is one-to-one (i. invertible, or “passes a horizontal line test”). These properties allow computers to handle numerical data more easily. Another popular, sigmoid function is g(z) = tan− 1 z

2 Section 4.

Determine both end behaviors for the given polynomials, i. determine the limits as x → ±∞.

(a) p(x) = 4x 3 − x 2 + 2x 9 lim x→−∞ p(x) = −∞ and lim x→∞ p(x) = ∞

(b) m(x) = 1 + 6x 3 − 7 x 5 + 10x 6 lim x→−∞ m(x) = ∞ and lim x→∞ m(x) = ∞

(d) y(x) = 5 − 9 x 51 + 10x 101 − 1000 x 102 lim x→−∞ y(x) = −∞ and lim x→∞ y(x) = −∞

Determine both end behaviors for the given rational functions, i. determine the limits as x → ±∞.

(a) p(x) =

4 x 3 − x 2 + 2x 9 1 + 6x 3 − 7 x 5 + 10x 6 Since the dominant terms are 2x 9 and 10x 6 , we find that lim x→−∞ p(x) = 1 5

x 3 = −∞

lim x→∞ p(x) = 1 5

x 3 = ∞

(b) m(x) = 1 + 6

x 3 − 7 x 5 + 10x 6 16 − x 7 Since the dominant terms are 10x 6 and −x 7 , we find that lim x→−∞ m(x) = − 10 /x = 0

xlim→∞ m(x) = − 10 /x = 0

x 2 − 3 x 7 + 3 − x 4 18 x 4 − 20 x 3 + 2 Since the dominant terms are − 3 x 7 and 18x 4 , we find that lim x→−∞ n(x) = −

1

6

x 3 = ∞

lim x→∞ n(x) = −

1

6

x 3 = −∞

(d) y(x) = 5

− 9 x 51 + 10x 101 − 1000 x 102 100 x 102 + 200x 101 − 41 Since the dominant terms are − 1000 x 102 and 100 x 102 , we find that lim x→−∞ y(x) = − 1000 /100 = − 10

xlim→∞ y(x) = − 1000 /100 = − 10

Gaussians (revisit): Previously, we introduced the Gaussian function below.

f (z) =

1 √

2 π e

− 0. 5 z 2

(a) What are the intercepts on the horizontal or vertical axes (if any)? Horizontal (z): none Vertical (y): (0, 1 /

√

2 π) (b) Evaluate lim z→∞ f (z) and lim z→−∞ f (z) Both are 0.

Sigmoid Function (revisit): Previously, we introduced the sigmoid function below.

f (z) =

1

1 + e−z

(a) What are the intercepts on the horizontal or vertical axes (if any)? Horizontal (z): none Vertical (y): (0, 1 /2) (b) Evaluate lim z→∞ f (z) and lim z→−∞ f (z) lim z→∞ f (z) = 1 and lim z→−∞ f (z) = 0

Consider the function f (x) = sin(

x) 5 x

.

(a) What is the slope of N (x) = sin(3x) at x = 0? N ′(x) = 2 cos(3x) giving N ′(0) = 3. (b) What is the slope of D(x) = 5x at x = 0? D′(x) = 5

Evaluate each limit. Use L’Hopital’s Rule if it applies (be sure to indicate indeterminate forms).

(a) lim x→ 0

ex − 1 sin x Indeterminant form: “0/0” and yields a limit of 1.

(b) lim x→ 1

ln x x − 1 Indeterminant form: “0/0” and yields a limit of 1.

(ln x) 3 x 2 Indeterminant form: “∞/∞” and yields a limit of 0.

Find the horizontal asymptotes of

f (x) = 2

x 3 + 5x 2 √ 9 x 6 − 1

lim x→∞ f (x) = lim x→∞

2 x 3 3 |x 3 |

= lim x→∞

2 3 = 2 3

lim x→−∞ f (x) = lim x→−∞

2 x 3 3 |x 3 |

= lim x→−∞

−

2 3 =

−

2

3

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Math 11-Worksheet 10 Sol

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 10 for Sections 4.5, 4.6, and 4.8 (Solutions) Spring 2023

1 Section 4.5

1. For the given functions, where is it concave up/down? Give your answers in interval notation. You

may use the work done from Worksheet 9!

(a) f(x) = x3+ 3x2−1

❼

f′′ (x) = 6x+ 6 and setting f′′ (x) = 0, we get x=−1. f(x) is concave up on [−1,∞) and

concave down on (−∞,−1].

(b) g(x) = √3x−2 sin x, on (0,2π)

❼

g′′ (x) = 2 sin xand setting g′′ (x) = 0, we get x=π(x= 0 is not in the interval). g(x) is

concave up on (0, π] and concave down on [π, 2π).

x+ 1

❼

h′′ (x) = 8

(x+ 1)3and setting h′′ (x) = 0, we find that there are no solutions, but still a critical

value at x=−1. h(x) is concave up on (−1,∞) and concave down on (−∞,−1).

2. Gaussians: (part 2) In Math 15, Math 18, and/or Psychology 10; you will encounter a Gaussian

curve—colloquially known as a “bell curve”. The standard, normal distribution is defined by the

function

f(z) = 1

√2πe−0.5z2

Familiarize yourself with the shape1of the function by analyzing the function for properties such as

the following:

(a) Over what intervals is the function concave up? ... concave down? Are there any points of

inflection? You may use the work done from Worksheet 9!

Here,

f′′ (z) = z2−1

√2πe−z2/2.

CU on (−∞,−1]∪[1,∞) and CD on [−1,1]. Points of inflection are (−1,1/√2eπ) and (1,1/√2eπ)

3. Sigmoid Function: (part 2) In a branch of computer science called Machine Learning, scientists

like to employ sigmoid functions to transform data.2Analyze the function

f(z) = 1

1 + e−z

for properties such as the following:

(a) Over what intervals is the function concave up? ... concave down? Are there any points of

inflection? You may use the work done from Worksheet 9!

Here,

f′′ (z) = −e−z(1 −e−z)

(1 + e−z)3.

CU on (−∞,0] and CD on [0,∞). Point of inflection is (0,1/2).

1Further (optional) reading: The reason for the coefficient 1

√2πis to ensure that the area between the curve and the horizontal

axis is equal to one. Considering that the bell curve is used in the study of probabilities, why is it important for that area to

be equal to one?

2Further (optional) reading: The goal of a sigmoid function is to map all possible inputs to a relatively small range of

numbers. Furthermore, a sigmoid function is one-to-one (i.e. invertible, or “passes a horizontal line test”). These properties

allow computers to handle numerical data more easily. Another popular, sigmoid function is g(z) = tan−1z

Math 11-Worksheet 10 Sol

Calculus I (MATH 011)

University of California, Merced

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1 Section 4.

√

8

1

√

√

√

1

.

2 Section 4.

1

6

1

6

1

√

√

1

.

2

3 = 2 3

−

2

3 =

−

2

3

Math 11-Worksheet 10 Sol

Course: Calculus I (MATH 011)

University: University of California, Merced

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