Question

Linear Approximation Let \(f\) be a function with \(f(0)=1\) and \(f^{\prime}(x)=\cos \left(x^{2}\right)\) (a) Find the linearization of \(f\) at \(x=0\) . (b) Estimate the value of \(f\) at \(x=0.1\) (c) Writing to Learn Do you think the actual value of \(f\) at \(x=0.1\) is greater than or less than the estimate in part (b)? Explain.

Short answer

The linearization of \(f\) at \(x=0\) is \(L(x) = 1+x\). The estimated value of \(f\) at \(x=0.1\) using the linearization is 1.1. However, since \(f'(x)=\cos(x^2)\) is a decreasing function for \(x>0\) which makes \(f\) be concave down, the actual value at \(x = 0.1\) should be less than the estimated value.

Step-by-step solution

Find the linearization of \(f\) at \(x=0\)

The linear approximation of a function at a certain point is given by the formula \(L(x) = f(a) + f'(a)(x-a)\), where \(a\) is the point around which we are approximating. In this case, \(a=0\), \(f(a)=f(0)=1\), and \(f'(a)=f'(0)=\cos\left((0)^2\right)=\cos(0)=1\). Therefore, the linearization of \(f\) at \(x=0\), denoted by \(L(x)\), is \(L(x) = 1 + 1 \cdot (x-0) = 1+x\).

Estimate the value of \(f\) at \(x=0.1\)

Now we can use \(L(x)\) to estimate the value of the function \(f\) at \(x=0.1\). Just plug \(x=0.1\) into \(L(x)\) resulting in \(L(0.1) = 1 + 0.1 = 1.1\). So, the estimated value of \(f(0.1)\) is 1.1.

Evaluate whether the actual value is greater than or lesser than the estimate

Given that \(f'(x)=\cos(x^2)\) is a decreasing function for \(x>0\), this means the function \(f\) is concave down on that interval. When a function is concave down, the linear approximation overestimates the exact values. So, the actual value of \(f(0.1)\) is predicted to be less than the estimated value of 1.1.

Key concepts

Linearization of a Function

Understanding the linearization of a function is a fundamental skill in calculus. Linearization is the process of approximating a function using a line, specifically the tangent line at a point. In a nutshell, it's like drawing a straight line that just touches ('tangents') the curve of your function at a particular point, then using that line as a stand-in for the curve nearby.

For the function f with a known value and derivative at a point, the linearized form, called L(x), gives us a neat linear function that is a close match to the original function around that chosen point. This is particularly useful because linear functions, being the simplest kind of polynomial, are much easier to work with than most other functions.

Derivative of a Function

The derivative of a function at a specific point quantifies the rate at which the function's value is changing at that point. If you picture the graph of a function, the derivative at a point is the slope of the tangent line at that point. The tangent line, therefore, is a straight line that skims the curve of the function without cutting through it.

In our exercise, the derivative is given as \(f'(x) = \cos(x^2)\). To find the linear approximation, or linearization, at a point, we need this derivative and a known function value. The derivative is like the engine of linearization - it determines how steep the tangent line will be and thus how accurate our linear model is around our point of interest.

Tangent Line Approximation

Tangent line approximation is a fancy way of saying we're making an educated guess about a function's values by looking at the values of its tangent line. It's based on the idea that for a small enough vicinity around the point of tangency, the function and its tangent line are almost identical. So, rather than dealing with potentially complex function behavior, you simplify the task by focusing on the straight line that's momentarily moving in sync with the function curve.

Tangent line approximation not only simplifies calculations but also opens the door to understanding how functions behave locally. In practice, by substituting the complex function with its tangent line around the point x=a, we can estimate the value of the function at nearby points with remarkable ease, just as we estimated f(0.1) using the linearization at x=0 in our exercise.

Concavity and Estimation Accuracy

The concavity of a function refers to the direction in which the function curves. If a function is curling up like the shape of a cup, it's concave up; if it curves down like a frown, it's concave down. Why is this important? Because concavity affects the accuracy of linear approximations. If your function is concave down and you're using a tangent line to approximate, as we see with \(f'(x)=\cos(x^2)\), the tangent line lies above the curve, leading to an overestimation.

On the other hand, concave up means we'd underestimate. This insight is a powerful part of error-checking in our estimates. The shape of the function can tell us whether our linear approximation will be slightly too high or too low, giving us an added layer of understanding about the behavior of the function near our point of approximation.