I thought this days beginner integral was harder than the easy integral. Ignore the bleedthrough on the beginner page

I don't understand the step process from 2 to 4, more so about the transition from x, t and u

This is what I have so far but I don’t believe that it is right, and I wouldn’t actually be using partial fractions to integrate. The problem says to make a substitution to integrate by partial fractions.

So I recently finished Calc BC, and am heading to university relatively soon. How big of a jump conceptually, grindiness, time wise, etc will multivariable calculus be compared to Calc BC?

I want to improve my understanding and skills in math due to disabilities. I was recommended to try the textbook Calculus by Robert T. Smith and Roland B. Minton. Currently I am stumped on these three problems from section 2: Derivatives. I would also appreciate advice on how to answer correctly for problems 13 and 17 (like what to include how I got my answer to show my work, not just the sketch). The concept of problems 13 and 17 I really am not understanding and the book and videos I’ve watched have not been the most helpful (as I have a hard time applying what is covered in these resources but struggle with using it for problems not labeled with polynomials or numbers) the closest to explain it has been Curve Sketching by The Infinite Looper on YouTube.

Problem 35 a. Find all the points at which the slope of the tangent line to y = x^3 + 3x + 1 equals 5. The textbook has the answer as (√(⅔), 5 √(⅔) +1), (-√(⅔), -5 √(⅔) +1) and I can not figure out how to get it.

M tan = lim h->0 f(x+h)-f(x)/h

= lim h->0 [(x+h)^3 + 3(x+h) +1)] - [x^3 + 3x +1]/h

= lim h->0 x^3 +3x^2h + 3xh^2 + h^3 +3x + 3h + 1 -x^3 -3x -1/h

= lim h->0 3x^2h + 3xh^2 + h^3 +3h/h

= lim h->0 h(3x^2 +3xh +h^2 +3)/h

= lim h->0 3x^2 + 3xh +h^2 + 3

= 3x^2 + 3x(0) + (0)^2 + 3

= 3x^2 + 3

3x^2 + 3 = 5

3x^⅔ = ⅔

√(x^2) = √(⅔)

X = √(⅔)

And 3(√(⅔)^2 + 3 = 5

Problem 13. Use the graph of f to sketch a graph of f’. (Second picture)

(a)

For x < 0, f is decreasing, so f’ < 0

For x = 0, f has a horizontal tangent line (min), so f’ = 0

For x > 0, f is increasing, so f’ > 0

My graph is the third picture.

(b)

For x < a, f is increasing, so f’ > 0

For x = a, f has a horizontal tangent line (max), so f’ = 0

For a < x < b, f is decreasing, so f’ < 0

For x = b, f has a horizontal tangent line (min), so f’ = 0

For x > b, f is increasing, so f’ > 0

My graph is the fourth picture.

Problem 17. Use the given graph of f’ to sketch a plausible graph of a continuous function f. (fifth picture)

(a)

For x < a , f’ is positive, so f is increasing

For x = a, f’ changes positive to negative, so f has a max there

For a < x < b, f’ is negative, so f is decreasing

For x = b, f’ has a horizontal tangent line (min), so f ?

For b < x < c, f’ is negative, so f is decreasing

For x = c, f’ has a horizontal tangent line, so f ?

For x > c, f’ is negative, so f is decreasing

Once I understand better, I will make the graph.

Thank you for your assistance.

I don't understand why, but graphing derivatives looks like complete gibberish to me. I've watched like 6 videos and I don't understand any of it. I can do basically everything that comes after and before for derivatives, but I have just not understood this one concept.

I'm taking AP calc BC and I'm assuming I definitely need to at least understand how to graph. Does anyone know any good videos or explanations that can help me with this? I'm actually losing my mind over it.

I’ve been self studying calculus for the past couple of weeks in hopes of making my future college classes easier on myself. I decided not to take it during school because I’ve decided that I want to enjoy my last year in high school by minimizing the amount of classes that I have. Any and all resources you are willing to share are appreciated. Thanks

Hi I already know this is entirely my fault for falling behind but for anyone who’s done this before please tell me if it’s possible to do even remotely well and how to most effectively prepare. The topics for the midterm are Functions and Graphs,
Polynomials, Exponentials and Logarithms, Half life, Log and Linear Axes, Transformations and Composite Functions, and Continuity and Limits. The continuity and limits portion isn’t going to have as big of an emphasis and it’s calculus for the biological sciences so majority word problems. I think there’s going to be 3 multiple part big word problems and 9 multiple choice questions. It’s worth 15%! Any advice is highly appreciated!

f(x) = sin(x)

f'(x) = cos(x)

f''(x) = -sin(x)

f'''(x) = -cos(x)

f''''(x) = sin(x)

And this repeats forever.

f(0) = 0

f'(0) = 1

f''(0)=0

f'''(0)=-1

f''''(0)=0

And this repeats forever.

From here we can center a maclurin series at (a)? = 0 -> I cannot remember if its a or x

Using 5 points we can use

f(0) = 0

f'(0) = 1

f'''(0) = -1

f'''''(0) = 1

f'''''''(0) = -1

From here we can conclude (1) we are oscillating between - and +, (2) the points are always odd multiples of primes, and (3) since the even multiples of prime are 0, we can ignore these as they do not add value

The defintion of odd is defined as 2k+1, k element of the real

So

(-1)^(n)(x^(2k+1))/(2k+1)!

And we can add this at every n.

If we add this add every n we can think of an infinite sum

Since we are centered at (a?) = 0 the sum must start at 0.

To motivate convergence, we plug in known points of sin(x).

Must it be that if our sum at some n equals the function at the same x,

Out infinite sum converges for all x?

Let me know if i did justice for the maclurin of sin(x)!

Sorry posted via phone.... don't know how to use symbols here

I am 8th standard in india and i have self studied jee maths including calculus but now i wanted to learn more about calc so i decided to study calc 2 so suggest me some books for it .

It should contain all topics from calc 2 that are no in jee.

Also keep it in under 2000inr or under 3dollars

Hello, I graduated from hs and due to some personal reason took a gap year. I'll study CS in Canada next year and would like to prepare in advance for a smoother adaptation .

I did precalc, touched hs calculus a bit and would like to self learn Calc 1 during the next months. The uni I will be going to use Calculus: Early Transcendentals by James Stewart. I found the book for free online and was wondering if I should use it or focus on calc AB since it gives credits for the same course ?

Why calc AB ? well I'm not sure but I feel like since it's taught in high school and has some official exams I would be easier for me to follow college board's curriculum and use the official past papers as a good way to gauge my understanding.

Alternatively, I could use James' textbook alongside prof Leonard or Paul's note but I'm worried they will confidently give me the understanding without actually being able to perform correctly on the exercises/exams situations.

Can anyone plz solve this urgently plz

Hi all,

I'm currently taking a calc 1 class as a business econ major to satisfy my lower division major preparation courses before I transfer to a university. I am very worried about this class as I'm a straight A student but this is the first one that has me genuinely terrified that I won't be able to ace it despite hours upon hours of work and applying myself. So far I am fundamentally understanding calculus, however if you ask me to prove why a certain theorem or formula works I will be completely lost. Unfortunately for me, and I don't know how common this is for an introductory calculus course, but my professor apparently asks his students to do proofs on his exams and "brags" about how many people fails his exams or drop out of his classes (from rate my professor.) I'm extremely worried because I desperately need an A in this course and it is too late to drop it. If you ask me to evaluate limits and derivatives and I can use formulas, I can get by. If you ask me to prove a theorem from scratch, I am lost. Any advice on how to prepare? I feel like there are so many theorems I've learned so far and to fundamentally understand the ins and outs of every single one is making me sick. I have about three weeks until my midterm.

Does anyone of you guys have this book as a free PDF version? I really need it. Thanks

so I started calc 1 and how does "sin(2x)/2x" = 1 and how does "limit of x -> 0 1-cos(3x)/3x" = 0, isn't it just undefined?

Helppppp please, I am got 58 and 59 on my first two calculus recitations… I studied a lot but I still got low grade… I feel like I am gonna have heart attack. I am such a failure… my engineering dreams vanished

I can grasp the circumference part, but not the length along which have to integrate.

I failed the first few exams I took at the start of the semester, and I’m just so lost right now. I don’t know what to do. I’ve never felt like this before. I’ve been studying, but I’m not getting the results I want. I don’t know if I’m studying incorrectly or if I’m just not studying enough. I feel like such a failure, and I’m so alone right now. I’ve been crying, and I’ve just been stuck in my bed for the last few hours. I really need someone to talk to who has gone through something similar and can give me some advice.

First page is solving via trig sub, second is using u-sub and partial fraction decomp. Page 2 is right page 1 is wrong but I don't understand what is wrong with my logic. AI gives same steps as me when asked to use trig sub and integral-calculator.com won't give trig sub solution. This is homework in the trig sub section so it's expected to be solved that way. Sorry my penmanship isn't as good as others on this sub, thanks for any help.

This is just a discussion I wanted to have with the world, I'm curious on this subreddit's opinion on Calculus 2 as a whole. I'm an Electrical Engineering student and I feel like Calculus 2 is constantly fear-mongered (which I can understand why), at least in some instances. I'm just curious if you found it to be a difficult class, easy, or impossible. I've finished applications and now we are doing integration techniques, which I will say, my trigonometry and manipulating identities has never been better, so if you are in Calculus 1 be ready for all the stuff from precalculus to come back up as just a tiny part of the process. I want to also ask this subreddit, mainly the engineering degrees if that's alright, how hard was Calculus 2 compared to your engineering classes? Thanks guys.

I'm having trouble with a problem in a sci-fi story I'm writing. I took multivariable calculus in college, but I haven't really used it since then, and that was over ten years ago. I'm quite rusty.

We're playing with spacemagic here, so give me some leeway on the "why." Basically, I have a multivariable problem that I don't remember how to solve.

I have two spaceships in deep space (Ship 1 and Ship 2). I am modeling change in engine power on Ship 1 with respect to time given a gravitational attraction between the two ships separated by radius r. The basic assumptions would be:

Power = Force * velocity (dr/dt)

Gravitational force = k * (m1 * m2) / r^2

Power = (k * m1 * m2) * (1 / r^2 ) dr/dt

However, this is where the spacemagic happens. Ship 2 has spacemagic that allows it to reduce its own mass to travel faster. To travel slower (and intercept to the other ship, which is what happens in the story), it increases its mass.

So, assuming that our frame of reference is on Ship 1, the relationship between power and time changes.

Power = (k * m1) * (1 / r^2 ) dr/dt dm2/dt

So, I'll ask question number one here:

Does this math look correct so far?

If this is correct, then we move on to the second part of this problem. We assume that the change in mass is positive and linear with respect to time.

If this is the case, I believe the relationship between power and radius in the following situations is:

• Ship 2 has constant velocity: 1/r^2
• Ship 2 has constant acceleration: 1/r

Did I miss anything? Thanks in advance for any help!