Given the function f(x) = 4 + 5x, calculate the following values: f(a) = f(a + h) : f(a + h) – f(a) | h
Given the function f(x) = 4 + 5x, calculate the following values: f(a) = f(a + h) : f(a + h) – f(a) | h
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![## Derivative Calculations
Given the function \( f(x) = 4 + 5x^2 \), calculate the following values:
1. \( f(a) = \)
- (Equation to find \( f(a) \), which represents the function evaluated at a point \( a \).)
2. \( f(a + h) = \)
- (Equation to find \( f(a + h) \), which represents the function evaluated at \( a + h \), where \( h \) is a small increment.)
3. \(\frac{f(a + h) - f(a)}{h} = \)
- (Expression to find the difference quotient, which is used to calculate the derivative of the function at point \( a \).)
**Explanation**:
- The function given is \( f(x) = 4 + 5x^2 \).
- To calculate \( f(a) \), you substitute \( a \) for \( x \) in the function.
- To calculate \( f(a + h) \), you substitute \( a + h \) for \( x \) in the function.
- To find the difference quotient, which approximates the derivative as \( h \) approaches 0, you use the expression \( \frac{f(a + h) - f(a)}{h} \).
### Sample Calculation Steps:
1. **Evaluate \( f(a) \)**:
\[
f(a) = 4 + 5a^2
\]
2. **Evaluate \( f(a + h) \)**:
\[
f(a + h) = 4 + 5(a + h)^2
\]
Expand and simplify:
\[
f(a + h) = 4 + 5(a^2 + 2ah + h^2) = 4 + 5a^2 + 10ah + 5h^2
\]
3. **Difference Quotient**:
\[
\frac{f(a + h) - f(a)}{h} = \frac{(4 + 5a^2 + 10ah + 5h^2) - (4 + 5a^2)}{h}
\]
Simplify:
\[
\frac{4 + 5a^2 + 10ah + 5h^2 -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e785cc8-9e88-4647-860a-d0a2baec2c71%2Ff8beeb99-7b0a-41d4-8aec-21c7ea5b54a5%2Fj4xfe6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Derivative Calculations
Given the function \( f(x) = 4 + 5x^2 \), calculate the following values:
1. \( f(a) = \)
- (Equation to find \( f(a) \), which represents the function evaluated at a point \( a \).)
2. \( f(a + h) = \)
- (Equation to find \( f(a + h) \), which represents the function evaluated at \( a + h \), where \( h \) is a small increment.)
3. \(\frac{f(a + h) - f(a)}{h} = \)
- (Expression to find the difference quotient, which is used to calculate the derivative of the function at point \( a \).)
**Explanation**:
- The function given is \( f(x) = 4 + 5x^2 \).
- To calculate \( f(a) \), you substitute \( a \) for \( x \) in the function.
- To calculate \( f(a + h) \), you substitute \( a + h \) for \( x \) in the function.
- To find the difference quotient, which approximates the derivative as \( h \) approaches 0, you use the expression \( \frac{f(a + h) - f(a)}{h} \).
### Sample Calculation Steps:
1. **Evaluate \( f(a) \)**:
\[
f(a) = 4 + 5a^2
\]
2. **Evaluate \( f(a + h) \)**:
\[
f(a + h) = 4 + 5(a + h)^2
\]
Expand and simplify:
\[
f(a + h) = 4 + 5(a^2 + 2ah + h^2) = 4 + 5a^2 + 10ah + 5h^2
\]
3. **Difference Quotient**:
\[
\frac{f(a + h) - f(a)}{h} = \frac{(4 + 5a^2 + 10ah + 5h^2) - (4 + 5a^2)}{h}
\]
Simplify:
\[
\frac{4 + 5a^2 + 10ah + 5h^2 -
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