Questions tagged [secant]
For questions about secant lines, which are lines that pass through two points on some curve.
62
questions
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2
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70
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Tangent lines equal secant line
Consider a continuous function $f$ with exactly three distinct points
$$
x_1(m) < x_2(m) < x_3(m)
$$
with slope $m$, meaning
$$
\frac{f(x_3(m))-f(x_1(m))}{x_3(m)-x_1(m)}=f'(x_3(m)) = f'(x_2(m)) =...
- 317
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154
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A notion of "differentiation" based on secant rather than tangent
Given a differentiable real function $f$, the derivative $f'(x)$ is the slope of the tangent to the graph of $f$ at $(x,f(x))$.
Suppose that, instead of the tangent, we look at the secant to the graph ...
- 10.8k
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1
answer
41
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Need Help for proofs in the Basel problem. [closed]
My teacher told me to prove that $$\tan ^{−1}
\frac {(1−𝑢)}{\sqrt{(1−𝑢^2)}}$$ leads to $$\tan^2(\theta) =\frac {(1 − u)}{(1+ u)}$$ and $$\sec^2(\theta) = \frac {2}{(1+u)}$$ using trigonometric ...
0
votes
1
answer
55
views
Difference of derivative and slope of secant line
Let $f \colon [x_0,x_1] \to \mathbb{R}$ be smooth and $|f^{\prime \prime}(x)|\leq L$ for some $L \in \mathbb{R}^+$. I want to have an estimation of the form
$$
\left| f^\prime(x) - \frac{f(x_1)-f(x_0)}...
- 55
0
votes
1
answer
46
views
How to calculate the value of $\log_b(x)$ using root finding secant method.
I am trying to calculate the value of $\log_b(x)$ for any $b\in(0, \infty)$ and any $x>0.$ I am supposed to do this only using basic arithmetic operations and exponentiation.
I know that in order ...
1
vote
1
answer
1k
views
Convergence of The Secant Method
I've been studying on some root finding techniques including The Bisection Method, False Position, The Secant Method and Newton-Raphson Method.
I've seen proof of convergence for all of these ...
- 71
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43
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Why is it that when the lattice points of the √x function are connected, the area between the secant line and the √x function is constant?
If you connect the lattice points of $f(x) = \sqrt{x}$ together through secant lines, you create a function $ g(x) = \left\lfloor \sqrt{x} \right\rfloor + \frac{x - \left\lfloor \sqrt{x} \right\rfloor^...
- 122
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votes
2
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Evaluate $\sum_{n\geq 0} \mbox{arccot}(n^2 + n + 1)$ [duplicate]
(This is a 1986 Putnam Challenge problem.)
First, note that
\begin{equation}
n^2 + n + 1 = \frac{n^3 - 1}{n - 1},
\end{equation}
which is the slope of the secant line through $f(x) = x^3$ at $x = 1$ ...
- 187
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1
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833
views
Secant method in Python
I have a function $f(x) = sin(2.5x)*e^x+0.5$. And I need to find the root in the interval [0,2]. But the problem is when I use the secant method I got the root, not in the interval [0,2].
I got $x= -...
-1
votes
1
answer
388
views
Determine the numbers a, b and c such that it satisfies the condition
I have a function $f(x) = x^3 + ax^2 + bx+ c$ and I need to solve for the numbers $a$, $b$, and $c$. The numbers need to satisfy the following condition:
The slope of the secant line defined by points ...
- 3
1
vote
2
answers
655
views
Prove when Instaneous Velocity is equal to Average Velocity with Constant Acceleration
Assume constant acceleration. It seems that average velocity over some time interval [t1, t2], will be equal to the instantaneous velocity at the midpoint t = 1/2[t1 + t2]. I'm wondering how you might ...
2
votes
0
answers
72
views
No lines meeting a curve in at least three distinct points implies no lines meeting a curve in three points counted with multiplicity?
Suppose $X\subset\Bbb P^3$ is a smooth projective curve over an algebraically closed field. Define a multisecant to be a line $L$ which intersects $X$ in at least three distinct points. If $X$ has no ...
- 2,840
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Name and/or generalization? "The slope of the secant through two points of a quadratic is the average of the slopes of the tangents at those points."
Does the following property of quadratic equations have a name? Is it generalized in some way? Or generalized to other functions?
Pick any two points on the graph of any quadratic. Draw a secant ...
- 55
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158
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Use secant method to approximate the (local) maximum of the function $f(x)=x^2 \cot{x}, x>0$
I'll be honest with you - as a designer, I'm not really someone who's into mathematics, but I need to solve one problem that's bugging me a lot. Can you help me to solve it? I can't find the solution ...
- 419
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3
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263
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Is $\int_{-\pi/2}^{\pi/2}\sec(x)$ bounded?
Is $\int_{-\pi/2}^{\pi/2}\sec(x)$ bounded?
It seems like it shouldn't be, since:
$$\int \sec(x) = \ln |\sec(x) + \tan(x)|+C$$
and $\sec(x)\to\infty$ as $x \to (\pi/2)^-$. But, I know integrals can ...
- 553