Calculus problems, where to begin?

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In summary: The gradient of a function at a point is equal to the rate of change of the function at that point. In this problem, you are asked to find the direction of the gradient of a function. So, you would use the gradient to find the direction of the gradient of the function. Hope this helps. In summary, the problem you are currently stuck on is a math problem that you are not sure how to begin, partially because you do not fully understand the problem, and partially because you are unsure if the level curves apply to the problem. You were wondering if anyone could help you out and someone responded. The person told you that the problem you are stuck on is a math problem that is constant on the
  • #1
Jeebus
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Hello! I have a math problem (mostly proofs) that I am stuck
on, partially because I do not know where to begin and partially
because I believe I don't even fully understand the problem. I
was wondering if any of you would be kind enough to show me what to
do? Thank you.

1. Suppose f(x,y) is differentiable for all (x,y), f(x,y)=17 on the
unit circle x^2+y^2=1 and grad f is never zero on the unit circle. For
any real number K, find a unit vector parallel to grad
f(cos(k),sin(k))...grad f stands for the gradient of f. But isn't it contradicting what its saying? It says f(x,y)=17 on the unit circle x^2+y^2. How the...?

I'm just supposing f(x,y) is differentiable for all (x,y), f(x,y)=17 on the unit circle x^2+y^2=1 and grad(f) is never zero on the unit circle(?) So you just find a unit vector parallel to grad f(cos(k),sin(k)), for k real, right?

PS- Do level curves apply to this problem?
 
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  • #2
Hello Jeebus,

You're being asked to find the direction of ∇f on the unit circle (k is just an angle). I think it's easier if we use polar coordinates (r,θ), and their corresponding unit vectors r and θ (don't know how to make the little hats yet). If we look at the gradient on the circle, and dot it with θ: θ dot ∇f, we get the rate of change of f in the θ direction. But what is the rate of change of f in the θ-direction on the unit circle? And what can you conclude about the direction of ∇f from this?

Hope this helps,
dhris
 
  • #3
dhris was giving hints that should help you but I got the impression that you really had no idea what was going on (and so need more than hints).

You ask "Do level curves apply to this problem?" Well, yes, of course. You are GIVEN that f(x,y)= 17 on the unit on the unit circle. The point is that f(x,y) is CONSTANT on that circle. The unit circle IS a level curve. Now, what is the relationship between level curves of a function and the gradient of the function at points on a level curve?
 

1. What is calculus?

Calculus is a branch of mathematics that deals with the study of change and motion. It is divided into two main branches, differential calculus and integral calculus, and is used to solve problems involving rates of change, optimization, and finding areas and volumes of irregular shapes.

2. How do I start solving a calculus problem?

The first step in solving a calculus problem is to carefully read and understand the given problem. Identify what is known and what is unknown, and determine which concepts and formulas are relevant to the problem. It is also important to draw diagrams or graphs to help visualize the problem.

3. What are some common techniques used in calculus problem solving?

Some common techniques used in calculus problem solving include finding derivatives, using the fundamental theorem of calculus, and setting up and solving integrals. It is also important to understand and apply the rules of differentiation and integration, as well as know how to use the chain rule and product/quotient rule.

4. What are some tips for success in solving calculus problems?

Practice and patience are key to success in solving calculus problems. Make sure to review and understand the fundamental concepts and formulas, and practice solving problems of varying difficulty levels. It is also helpful to break down complex problems into smaller, more manageable parts, and to check your work for errors.

5. Where can I find additional resources for practicing calculus problems?

There are many online and offline resources available for practicing calculus problems, such as textbooks, practice workbooks, and online problem sets and quizzes. Additionally, many universities and educational websites offer free resources and tutorials on calculus problem solving. It is also beneficial to seek help from a tutor or join a study group to get additional support and guidance.

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