Question 2. A function f(x) satisfies f(x+h)-f(x)=x+h+z² · h² + 4h³ for all z and h. Use Fermat's method of adequality (difference quotient) to find of the tangent line when z = 2. (a) z+z²
Question 2. A function f(x) satisfies f(x+h)-f(x)=x+h+z² · h² + 4h³ for all z and h. Use Fermat's method of adequality (difference quotient) to find of the tangent line when z = 2. (a) z+z²
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Mathematics: Exam Preparation
#### Detailed Questions and Solutions
**Question 2:**
A function \( f(x) \) satisfies
\[ f(x + h) - f(x) = x \cdot h + x^2 \cdot h^2 + 4h^3 \]
for all \( x \) and \( h \). Use Fermat’s method of adequation (difference quotient) to find the slope of the tangent line when \( x = 2 \).
Options:
(a) \( x + x^2 \)
(b) 2
(c) 6
(d) \( 2h + 4h^2 + 4h^3 \)
-----
**Question 3:**
Compute the sum of divisors of \( 2^5 \cdot m \), where \( m \) is an odd number. (Recall that \( s(N) = \sum \text{ of divisors of } N \).)
Options:
(a) \( 31 \cdot s(m) \)
(b) \( 63 \cdot s(m) \)
(c) \( 64 \cdot s(m) \)
(d) It is impossible to say without more information.
-----
**Question 4:**
Compute the number of divisors of \( 4^3 \cdot m \) where \( m \) is an odd number. (Recall that \( \tau(N) = \text{number of divisors of } N \).)
Options:
(a) \( 6 \cdot \tau(m) \)
(b) \( 10 \cdot \tau(m) \)
(c) \( 11 \cdot \tau(m) \)
(d) It is impossible to say without more information.
-----
**Question 5:**
If \( p > 3 \) is a prime, what’s the remainder when \( (1 + p)^p + 2 \) is divided by \( p \)?
Options:
(a) 0
(b) 1
(c) 2
(d) 3
-----
**Question 6:**
What’s the square/square-free factorization of 520? Recall that this means \( 520 = s \cdot r^2 \) where \( s \) is square-free.
Options:
(a) \( s = 2, r = 4 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96c79fcf-e51e-4a22-9a17-92d75ad76ddb%2F7881c33b-1414-4370-8798-c99e7bfb0b19%2Fmmhbfmq_processed.png&w=3840&q=75)
Transcribed Image Text:### Mathematics: Exam Preparation
#### Detailed Questions and Solutions
**Question 2:**
A function \( f(x) \) satisfies
\[ f(x + h) - f(x) = x \cdot h + x^2 \cdot h^2 + 4h^3 \]
for all \( x \) and \( h \). Use Fermat’s method of adequation (difference quotient) to find the slope of the tangent line when \( x = 2 \).
Options:
(a) \( x + x^2 \)
(b) 2
(c) 6
(d) \( 2h + 4h^2 + 4h^3 \)
-----
**Question 3:**
Compute the sum of divisors of \( 2^5 \cdot m \), where \( m \) is an odd number. (Recall that \( s(N) = \sum \text{ of divisors of } N \).)
Options:
(a) \( 31 \cdot s(m) \)
(b) \( 63 \cdot s(m) \)
(c) \( 64 \cdot s(m) \)
(d) It is impossible to say without more information.
-----
**Question 4:**
Compute the number of divisors of \( 4^3 \cdot m \) where \( m \) is an odd number. (Recall that \( \tau(N) = \text{number of divisors of } N \).)
Options:
(a) \( 6 \cdot \tau(m) \)
(b) \( 10 \cdot \tau(m) \)
(c) \( 11 \cdot \tau(m) \)
(d) It is impossible to say without more information.
-----
**Question 5:**
If \( p > 3 \) is a prime, what’s the remainder when \( (1 + p)^p + 2 \) is divided by \( p \)?
Options:
(a) 0
(b) 1
(c) 2
(d) 3
-----
**Question 6:**
What’s the square/square-free factorization of 520? Recall that this means \( 520 = s \cdot r^2 \) where \( s \) is square-free.
Options:
(a) \( s = 2, r = 4 \
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