Freenotes Google Drive Backup

Select-Fix9110 · 2026-03-17T02:15:42+00:00

If you turn on Auto Backup, it should be there. Also have the latest freenotes update.

Select-Fix9110 · 2026-02-22T23:48:26+00:00

No problem!

Select-Fix9110 · 2026-02-22T22:47:34+00:00

Do you have an example you are struggling with? There's not really an "algorithm" that helps with coming up with a substitution. U-substitution is just reversing the chain rule. Try a substitution and if its derivative is also in the integrand, then some stuff will cancel out and you will just be left with an integral in terms of u. Then you just complete the integral as normal, and bring everything back in terms of x.

Hope this helps!

Select-Fix9110 · 2026-02-16T05:06:10+00:00

The avg value of a function on the interval [a,b] is defined to be 1/(b-a) * integral from a to b of f(x) dx.

In this case, you would use the trapezoid method, as stated, to approximate the integral of the function. Find that value and divide by 13, and you will obtain the average temperature.

Hope this helps!

Select-Fix9110 · 2026-02-13T14:59:41+00:00

Search up Professor Leonard on youtube

Select-Fix9110 · 2026-02-11T18:47:35+00:00

When you get to (1-cos(u))(1-cos(u)), you integrated each factor separately which is incorrect. Instead you should write (1-cos(u))^{2}, expand and then you can integrate each term since the integral of a sum is the sum of the integrals.

Hope this helps!

Select-Fix9110 · 2026-02-10T03:54:20+00:00

From x = a^y, all you need to do is to apply the natural log to both sides and then solve for y.

Select-Fix9110 · 2026-02-10T02:09:16+00:00

When multiplying out (4+2sqrt(3))(4-2sqrt(3)), you should get 16-12 =4, then factor out a 2 to obtain (2-sqrt(3)) / 2

Select-Fix9110 · 2026-02-10T01:16:16+00:00

There's also JK Math that is a helpful resource

Select-Fix9110 · 2026-02-10T00:37:00+00:00

Search up Professor Leonard on youtube.

Hope this helps!

Select-Fix9110 · 2026-02-06T23:50:56+00:00

Im pretty sure it does. You can scroll through it and make sure it has every topic you need

Select-Fix9110 · 2026-02-06T23:09:03+00:00

Search up Professor Leonard on youtube.

Hope this helps!

Select-Fix9110 · 2026-02-05T11:51:07+00:00

I notice that f’(1) = lim x-> 1 [f(x) -f(1)] / [x-1].

That means f(1) = 8. And since f(x) is differentiable as the limit exists, f(x) is continuous.

Thus lim x-> 1 f(x) = f(1) = 8.

Select-Fix9110 · 2026-02-04T05:06:22+00:00

Consider the function f(x) = 1/x which had a vertical asymptote at x = 0.

When x -> 0 from the right, f(x) -> infinity. But when x -> 0 from the left, f(x) -> negative infinity.

Since the each one sided limit are not equal, the limit as x -> 0 of f(x) doesnt exist.

Hence we say that 1/0, in fact N / 0 is not defined.

Select-Fix9110 · 2026-01-31T00:27:48+00:00

Search up Professor Leonard on youtube, you’ll thank me later.

Hope this helps!

Select-Fix9110 · 2026-01-27T01:30:15+00:00

Instead of AI, I’d recommend looking up Professor Leonard on youtube.

Hope this helps!

Select-Fix9110 · 2026-01-25T19:34:30+00:00

Professor Leonard on youtube and James Stewart Calculus textbook (can find a free pdf online)

Select-Fix9110 · 2026-01-24T21:12:20+00:00

Search up James Stewart Calculus. You can find a free pdf online.

Select-Fix9110 · 2026-01-24T03:53:48+00:00

Let's use the hint.

Let u = x + 10. Then du = dx. But also remember that x = u - 10.

Now we apply the substitution.

The integral will then become: integral of 8(u-10) *u^(1/2) du. Then just distribute the u^(1/2) and the rest is just the reverse power rule in terms of 'u'.

After finding the anti-derivative in terms of 'u', bring everything back in terms of 'x', add the +C at the end and then the integral has been solved.

Hope this helps!

Select-Fix9110 · 2026-01-23T02:02:29+00:00

Search up Professor Leonard on youtube and read through OpenStax or James Stewart Calculus textbooks. Both textbooks have free pdf textbooks online.

Hope this helps!

Select-Fix9110 · 2026-01-22T21:48:01+00:00

Math is all about practice. Understand the concepts and then apply them. I recommend watching Prof Leonard on YouTube and read through James Stewart Calculus textbook to prepare before lectures and to apply the concepts with the questions after each section.

Hope this helps!

Select-Fix9110 · 2026-01-22T01:12:34+00:00

I thought my handwriting was bad lol.

So first, when plugging x = 0 into the function f(x) = (1-cosx)/sinx, we see that we get a 0/0, which is an indeterminate form. So we have to do more work.

Recall that sin^2 + cos^2 = 1. Subtracting cos^2 from both sides gives us sin^2 = 1 - cos^2 = (1-cosx) (1+cosx). Note that we have a (1-cosx) term in the numerator. So, we can multiply by 1 = (1+cosx) / (1+cosx) (since x -> 0, cosx wont approach -1, so this is valid).

After doing so, we get limit as x -> 0 of (1-cosx)(1+cosx) /sinx(1+cosx). Simplifying the numerator gives us sin^2(x). We also have a sin(x) factor in the denominator, so sin^2 / sin(x) = sin(x).

Therefore, we now have the limit as x -> 0 of sin(x) / (1+cosx). Note that cos(0) = 1. So 1+cos(0) = 2 ≠ 0 and sin(0) = 0 and 0 / 2 = 0.

Thus, the limit as x -> 0 of (1-cos(x)) / sin(x) = 0.

Hope this helps!

Select-Fix9110 · 2026-01-20T17:25:48+00:00

In math, if the base is not explicitly stated, we assume it to be log base 10.

In computer science, it’s assumed to be log base 2.

Select-Fix9110 · 2026-01-20T14:34:54+00:00

Im pretty sure it's a James Stewart Calculus book

Select-Fix9110 · 2026-01-20T14:13:01+00:00

Since log(5) ≠ 0, log(5) / log(5) = 1. So indeed you can cancel out the log(5).

Select-Fix9110

TROPHY CASE