Suppose that f(1) = 10 and the graph of the derivative of f is shown below. 6 5 4 y = f '(x) 2 1 3 Use linear approximation to estimate f(1.2). 2. 3.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![### Topic: Linear Approximation in Calculus
#### Problem Statement:
Suppose that \( f(1) = 10 \) and the graph of the derivative of \( f \) is shown below.
#### Graph Description:
The graph displayed is of the function \( y = f'(x) \), which represents the derivative of \( f(x) \). The graph is depicted on a coordinate plane where:
- The x-axis ranges from 0 to 3.
- The y-axis ranges from 0 to 6.
The curve starts from the origin (0,0), steeply rises as \( x \) increases from 0 to 1, and then gradually levels off as \( x \) continues to increase. The curve approaches \( y = 5 \) as \( x \) approaches 3.
#### Task:
Use linear approximation to estimate \( f(1.2) \).
Input field: [__________]
### Explanation:
To solve this problem using linear approximation, we use the formula:
\[
L(x) = f(a) + f'(a) (x - a)
\]
where \( L(x) \) is the linear approximation of \( f(x) \) at \( x \). Given that \( f(1) = 10 \), we identify \( a = 1 \). We also need the value of \( f'(1) \), which can be read off from the graph.
Using the graph, when \( x = 1 \), \( y = f'(1) \) is approximately equal to 5 (as inferred from the graph).
Substituting into the linear approximation formula, we estimate \( f(1.2) \):
\[
f(1.2) \approx f(1) + f'(1)(1.2 - 1)
\]
\[
f(1.2) \approx 10 + 5(0.2)
\]
\[
f(1.2) \approx 10 + 1
\]
\[
f(1.2) \approx 11
\]
Hence, \( f(1.2) \) is approximately 11.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ca79b19-dfd2-4956-96e7-5567974ae84a%2F153ae99e-6613-4954-a41a-d9912099c5ed%2Fs39x17q_processed.png&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
