# Questions tagged [linear-approximation]

For questions about linear approximations, $f(x) \approx f(a)+f'(a)(x-a)$ for $x$ around $a$.

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### How can we prove that some function, .e.g., the hyperbolic tangent function, tends to be linear around zero?

I have found this interesting answer about increasing the linear range of the hyperbolic tangent function. Now, I am looking for a proof (or at least have a reference from literature if it shows up to ...
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### Why does taking the tangent line improve the approximation in Newton's method?

I have gained a comprehension of the operational process through the discussion located at Why does Newton's method work?. Nevertheless, there is one aspect that remains unclear to me. To initiate,...
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### Jacobian of azimuth and elevation angles with respect to unit vector

We know that azimuth ($\theta$) and elevation ($\phi$) angles can represent a unit vector as $\mathbf{e}=\begin{bmatrix}\cos\theta\cos\phi \\ \sin\theta\cos\phi \\ \sin\phi\end{bmatrix}$. It is easy ...
1 vote
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### How to get exact value from multiple approximate values

So I was on amazon looking for measurement cups and stumbled upon this set that had a measuring table for converting between one measurement and another. Here is the link for reference Amazon ...
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### Nonlocal linear approximation to nonlinear ordinary different equation

Suppose I have a nonlinear ordinary differential equation, in several variables, with a stated initial condition. How would I go about finding a nonlocal linear approximation? What is known about such ...
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### How to calculate the error of a taylor polynomial in multivariable

So I've tried googling but I am not finding any information about the error in mulitvariable, everything out there is about single variable. In my prof powerpoint it says for quadratic approximation ...
1 vote
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### Linear approximation in multivariable

Find the tangenplane at (1,2,3) $f (x, y ) = \frac{x^2y}{y − 1} + 1$, then approximate the value of the function at f(1.2, 2.3) I partial differentiated and got $f_x$ = 4 and $f_y=-1$ then I used the ...
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### Accuracy of linear approximation

I am working on Calculus I by Marsden and Weinstein and this is Exercise 43 from Section 1.6 Let $g(x)=-4x^2+8x+13$. Show that the linear approximation to $g(3+\Delta x)$ always gives an answer which ...
433 views

### Contradiction in derivatives as linear approximations

From the definition of a derivative, we have that $$f'(a) = \lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$$ or $$\lim\limits_{x\to a}f'(x) = \lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$$ This leads me to ...
1 vote
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### Complex Differentiability at a Point Equivalent to Having an Affine Part

I’m trying to prove that a function $f: \mathbb C \rightarrow\mathbb C$ continuous at some point $a$ has affine part $Az+B\overline z+C$ if and only if it is differentiable at $a$. Note: The affine ...
1 vote
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### What does a 'good' approximation mean?

I am a graduate student and currently studying functions of several variables.I am mainly following Paliogiannis and Moskowitz.When they are introducing differentiability for function of several ...
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1 vote
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### Using the tangent line of a value to approximate a value

Good Day, For some reason, my brain is failing to craft a reasonable solution to this equation. Question: Use the tangent line of $f(x) = \sqrt x$ at $a=16$ to approximate $\sqrt 17$ I believe we are ...
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### Approximating $\ln(1+2x) = 2x$ using linear approximation

The question is as follows: Use the linear approximation formula $$\boxed{f(x+\Delta x) \approx f(x) +f'(x)\Delta x}$$ to show that $\ln(2x+1) \approx 2x$ for small values of $x$. I am having ...
1 vote
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### Efficiently approximate Ax=b if you can choose values for A

Here is the situation. You have a black box x, whose values are not known to you, but you can choose values for A and generate the corresponding b value using Ax=b. There is some noise in the system ...
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### What is the reason for this step in proving the Borsuk-Ulam Theorem (by triangulation)

I'm reading Using the Borsuk-Ulam Theorem, which presents several proofs of the theorem. Since this is a book for combinatorialists, the first involves a triangulation $\mathsf{T}$ of the $n$-...
### Why does the linear approximation of $f \circ g$ near $a$ imply $f$ gets linearized by $g$ for a small enough neighborhood of $g$ near $a$?
I've noticed that, if we input the linear approximation of $g$ near $a$ into $f$, we get a decent approximation (althought non-linear) of the linear approximation of $f \circ g$ near $a$. In order to ...