Vector Calc Partial Deriv & Gradient?
Hi (a) Since from (3, 1) to (3, 2), x doesn't change at all, we can do: (f(3, 2) - f(3, 1))/(2 - 1) This evaluates to: (f(3, 2) - f(3, 1))/(2 - 1) = (f(3, 2) - f(3, 1))/1 = f(3, 2) - f(3, 1) = (3^2 + ln(2)) - (3^2 + ln(1)) = (9 + ln(2)) - (9 + 0) = 9 + ln(2) - 9 = ln(2) (≈ 0.693) So the average rate of change is ln(2) (b) Here, they're asking for the directional derivative of f at the point (3, 1) along a vector <3 - 3, 2 - 1> = <0, 1>. The directional derivative can be found by: D_u [f(x, y)] = <∂f/∂x, ∂f/∂y>•u Where u is the unit vector in the direction that you're headed. Here, u = <0, 1>. We can find <∂f/∂x, ∂f/∂y> to be (just treat x or y as constants): f(x, y) = x^2 + ln(y) ∂f/∂x = 2x ∂f/∂y = 1/y <∂f/∂x, ∂f/∂y> = <2x, 1/y> At the point (3, 1), <2x, 1/y> = <6, 1>. Take the dot product of that and u = <0, 1>. You get: 6*0 + 1*1 = 0 + 1 = 1 So the instantaneous rate of change of f is 1. I hope this helps!
Partial derivative problem?
-2ySin(x) - 3(x^2)/y Treat y as a constant to find df/dx = -(y^2)sin(x) - 3(x^2)ln(y) (partial d's, not normal d's) Then treat x as a constant in the above to find the result at the top.
How to find a tangent line (Calc III)?
I am having some trouble with the following problem: Find a parametric representation of the tangent line to the curve of intersection of the following surfaces x^2 + y^2 + z^2 = 49 and x^2 + y^2 = 13 at the point (3,2,-6). I really do not know whether when finding normal vectors at the point what the z component would be for the second equation. Would it be 0, as the partial derivative of f(x,y,z) with respect to z is 0, or would it be -6, because z is not included in the surface?
Multivariable calc- cylindrical coordinates?
2) using cylindrical coordinates: knowing that: x = r cos(θ) & y = r sin(θ) & r^2 = x^2 + y^2 x^2 + y^2 =16 =====> r^2 = 16 ====> r = 4 θ will be from 0 to 2π to let it rotate the whole region. 0 ≤ θ ≤ 2π ∫ 0 ≤ r ≤ 4 ∫ -5 ≤ z ≤ 4 ∫ √(x^2 + y^2) dz r dr dθ 0 ≤ θ ≤ 2π ∫ 0 ≤ r ≤ 4 ∫ -5 ≤ z ≤ 4 ∫ √[ r^2] dz r dr dθ 0 ≤ θ ≤ 2π ∫ 0 ≤ r ≤ 4 ∫ -5 ≤ z ≤ 4 ∫ r dz r dr dθ 0 ≤ θ ≤ 2π ∫ 0 ≤ r ≤ 4 ∫ -5 ≤ z ≤ 4 ∫ r^2 dz dr dθ 4 ∫ r^2 dz -5 .....4 r^2 z ] = r^2 * [ 4 - -5 ] ====> 9r^2 .....-5 4 ∫ 9r^2 dr 0 . . . . . . .4 (9/3) r^3 ] = 3 * [ 4^3 - 0 ] = 3 * 64 = 192 . . . . . .0 2π ∫ 192 dθ ====> 192 * 2π ≈ 1,206.37 unit^3 0 ---------------------------------------... using cylindrical coordinates: x = r cos(θ) & y = r sin(θ) & r^2 = x^2 + y^2 z^2=4x^2 +4y^2 =====> z= +/- √(4x^2 +4y^2) ====> z= +/- √(4r^2) ===> +/- 2r x^2 + y^2 = 1 ====> r^2 = 1 ===> r = 1 θ will be from 0 to 2π to let it rotate the whole region. 0 ≤ θ ≤ 2π ∫ 0 ≤ r ≤ 1 ∫ 0 ≤ z ≤ 2r ∫ √[ ((r cos(θ))^2) ] dz r dr dθ 0 ≤ θ ≤ 2π ∫ 0 ≤ r ≤ 1 ∫ 0 ≤ z ≤ 2r ∫ r cos(θ) dz r dr dθ 0 ≤ θ ≤ 2π ∫ 0 ≤ r ≤ 1 ∫ 0 ≤ z ≤ 2r ∫ r^2 cos(θ) dz dr dθ 2r ∫ r^2 cos(θ) dz 0 ..... . .. . . . 2r r^2 cos(θ) z ] = r^2 cos(θ) * [ 2r - 0 ] ====> 2r^3 cos(θ) ..... . . .. . ..0 1 ∫ 2r^3 cos(θ) dr 0 . . . . . . . . . . .. .1 (2/4) r^4 cos(θ) ] = (1/2) cos(θ) [ 1^4 - 0 ] = (1/2) cos(θ) . . . . . . . .. . . ..0 2π ∫ (1/2) cos(θ) dθ 0 . . . . . . .2π (1/2) sin(θ) ] = 0 . . . . . . . .0 ( because of the question which is sq.rt. (x^2) but if we don't have ba square root then we would have a number )
If you have taught or taken Calculus III (Multivariable Calc)?
Hey college professors, can you give me advice on my midterm? 10 points for in-depth answers? The course is Multivariable Calculus (Calculus III). Our professor is allowing us to have one sheet of 8 x 11in paper with notes with anything we want on it. What should I put? The midterm will be on the following topics. 1. Vectors and Geometry: Vectors in the plane and space, Dot Product, Cross Product. Lines, planes, spheres, Cylinders, quadric surface. Polar, Cylindrical, and Spherical coordinates. 2. Vector Valued Functions: Space Curves differentiation, integration and motion in space. Arclength and unit tangent vectors. Acceleration on space curves 3. Multivariable Calc: Domains, representations (level curves/contour plots), limits and continuity of scalar valued functions of several variables. Partial derivatives, Partial Differentials (Laplace's equations, not Laplace Multipliers) In addition, my prof told us that there is at least going to be one proof question for the Squeeze theorem of a limit or a proof for a derivative property for vectors. He also said there will be a few true or false questions and no M/C
Should linear algebra be taken before calculus 3?
You probably mean Calculus 3 in the sense of multivariable calculus. The answer is “no”, but once you do learn linear algebra, you need to come back and relearn the entire subject in that light. The class that does this is usually some combination of Real Analysis and Differential Topology, which unless you’re a math major you probably won’t take.Multivariable calculus really needs linear algebra to make sense. The reason is simply that the derivative of a function f (called its “differential” in this context) is the best approximation of f by a linear transformation. This is a fancy way of saying that the derivative is the slope of the tangent line, with both geometric terms generalized to higher dimensions. The major results of multivariable differentiation look quite nice in terms of linear transformations: for instance, the chain rule says more or less that differentiation commutes (changes order) with composition of functions; compare that to the lengthy sums-of-products expressions involving partial derivatives that don’t exactly cancel like fractions, which is the version you’ll see without linear algebra.There are so many other ways that linear algebra gives structure to multivariable calculus. It enables the statement of the generalized Stokes theorem, for instance. As part of that, it also explains how the dot and cross products come to be, how they are related, and why the regular Stokes theorem has one of them and the divergence theorem has the other one, even though they are otherwise very similar in their general structure.Basically, a first course of multivariable calculus is going to be a lot of unjustifiable assertions and messy computations, focusing on the things you can do without linear algebra: computing derivatives and integrals. A good pairing with it is not linear algebra but actually physics. Basically all the interesting ideas and results are physically motivated. This is an excellent way to gain insight into both subjects; this insight is complementary to that obtained from linear algebra.In a sentence, with respect to multivariable calculus, linear algebra is “why the how” and physics is “why the what”. It’s a course that actually leads pretty far into undergraduate mathematics if you go down the rabbithole.
Can someone explain the gradient to me? Calc III?
You mean gradient as in vector calculus? You can imagine a function f(x,y) of two variables as representing a surface, if you think of f as the height above some zero level. The gradient is a vector, with components partial df/dx and partial df/dy. At each point (x,y) on the surface the gradient vector points in the steepest direction "up hill". The magnitude of the vector is bigger when the slope of the "hill" is steeper. If you are at a place where the surface is "horizontal" (like a maximum or minimum point) the gradient vector is zero. If also turns out that if you imagine a contour map drawn on the surface, the gradient vector is always at right angles to the contour passing through the point. That is fairly obvious, since the gradient vector is the "steepest" direction up hill at every point. The gradient-and-contour-are-at-right-angles relationship is a very important idea when solving problems involving minimizing or maximizing functions of several variables. For functions of more than two variables, there isn't a simple way to visualise what the gradient means. You just have to accept that the math works the same way as it does for two variables. For example the "right angles" relationship can be written mathematically as "the scalar product of two vectors is zero" and that remains true for more than 2 variables. Hope that helps
How do you derive a function?
Firstly, the actual term, (Proper Calculus Jargon) is DIFFERENTIATE not derive.In order to differentiate a function, one must utilize differentiate techniques. This is usually taught in high school in Calculus AB in Single Variable Calculus.Some examples of differentiation techniques include….Power RuleBest used for a function that is raised to a power. For example a quadratic or cubic or a negatively raised function.Quotient RuleBest used for a rational function. When there are stuff on top and bottom. “Division”. This is quite useful actually.One can use the quotient rule for… f(x)=(3x^3)/(2x+3)Product RuleBest used for stuff that is multiplied. Two parts that are stuck together then this product rule is great!One can use the product rule for… f(x)=x^3(sin(x))Chain RuleA more advanced technique. Great for a function that has another one in it. It is tricky and can be a place of error for beginner calculus students.An example: f(x)=(3x+2)^(2)In order to differentiate the function, one must use the chain rule. This is by bringing the exponent 2 to the front, subtracting 1 from the exponent. Then MUST MULTIPLY BY THE DERIVATIVE OF THE INSIDE. This is 3.In the end, the derivative of that function, f(x) would be 2*3(3x+2) = 6(3x+2).Logarithmic DifferentiationMany techniques such as ones involving natural logs, standard base 10 logs, etc.Exponential DifferentiationMany rules which depends on the base, etc. Such as e, a constant.Trig Differentiation (Not hyperbolic or Inverse) Just good ol Sin, Cos, Tan, Sec, Csc, CotAgain, the 6 trig can be differentiated. There is some nice charts out there.Inverse trig Differentiation (Sin^-1, Cos^-1, etc)Again, refer to textbook or google!These differentation techniques will be very useful in BC Calc topics, such as getting the Taylor and Maclaurin series of a function!HAVE FUN! I certainly am!These are all the commonly used ones in AP Calculus. I am currently a sophomore taking it!Cheers! Anthony Hui
Calculus 3 final advice?
Hey everyone ... So this semester I had calculus 3 at 8 o clock in the morning (had no choice) which resulted in me missing many many classes because I couldn't hear my alarm .. I know this is irresponsible etc .. Because I missed many classes I have no idea about a lot of stuff so far .. I have 1 day to study for it because I had been studying so far for my other classes (in which I get Bs ..). I honestly just want to pass the course .. I am doing so bad that in order to pass I need to get 70% in the final (I had 75% average in quizzes that are 40% of the grade but i missed one accidentally which screwed me up ... i couldn't complete reports that counted for 10% and online homework on saturday nights was something i forgot many times ... also counted for 10%). Anything I should focus on? Some stuff that will definitely be on the exam? Some things that definitely won't? I can do it via copying but I don't really want to ...
How can I handle AP Physics C, AP Calc BC, Honors English 2, AP Gov, Spanish 3, and AP Comp Sci as a sophomore in high school?
Wow, we’re taking sort of the same courses in sophomore year, high five.I’m taking Honors Chem, AP Calc BC, English 10, APWH, Spanish 2, and AP Comp Sci A and self studying AP Physics C. I’m guessing we have similar interests (mathematical and sciency topics) based on the courses we’re taking, so hopefully I could relate to you better.I’d assume you have a pretty stable calculus background since you’re taking Physics C and Calc BC. BC is basically the applications of AB, so it’s not hard. As for AP Physics C, you should be able to learn the concepts pretty quickly, though I’ve noticed there is the use of line integrals and maybe surface integrals (from Multivariable Calculus, or calc III after BC) when learning about Faraday’s Law, and a few other electric field concepts. You should probably get familiarized with some basic multivariable calculus content (partial derivatives, line integrals, and surface integrals) prior to starting that section.I’ve never taken AP Gov before, but I’ve heard it is pretty easy, in fact it is the only required AP at my school which only requires 2 AP classes to graduate. That should be pretty easy if you pay attention in the class.I’m assuming you’re taking AP Comp Sci A, which is the one where you actually have to learn a programming language, Java. Get familiar with Java, and certain concepts, such as binary, octal, hexadecimal, and sorting algorithms (bubble, selection, insertion, etc). That’s basically all you need for that class.The other classes should be very easy, so I’m assuming you can handle those.Good luck!