What is Differential calculus: Definition and 73 Discussions
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.
Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.
Differentiation has applications in nearly all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration. The derivative of the momentum of a body with respect to time equals the force applied to the body; rearranging this derivative statement leads to the famous F = ma equation associated with Newton's second law of motion. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories.
Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra.
Homework Statement
Use partial differential calculus to show that if 3 quantities p, V, T are related to each other by some
unknown but smooth (which means all derivatives are well defined) equation of state f (P, V, T ) = 0. Then the
partial derivatives must satisfy the relation∂p/∂T = - (∂V...
Homework Statement
I don't understand how this can be written like this:
\frac{1}{(2x+1)^{1/2}}-\frac{1}{(2x+1)^{3/2}}=\frac{2x+1-1}{(2x+1)^{3/2}}=\frac{2x}{(2x+1)^{3/2}}
What's the rule which makes this possible and explain please.
Homework Statement
A ship moving at 8 mi per hour, Sails W for 2 hours, then turns N 30 E. A search light, placed at the starting point, follows the ship. Find how fast the light is rotating, (a) 3 hours after the start; (b) just after the turn.
Homework Equations
The Attempt at...
If I'm lucky enough to pass Precalculus this semester, I'll be taking Calculus 1 next semester. Just out of curiosity, what is the difference between differential and integral calculus? Also, how does the two of them relate to upper-level mathematics (e.i: Number Theory, Vector Calculus, Real...
HI guys, I just started university and I have no idea what is going on in my lectures. My lecturer doesn't explain anything well and he is also the bloke who takes my tutorials. Lose - lose situation! Anyways I am trying to learn this myself, but I am stumped on this question:
Homework...
Homework Statement
A projectile is fired straight upward with a velosict of 400m/s. From physics, its distance above the ground after t seconds is: s(t) = -16t^2 + 400t
A Find the time and velocity at which the projectile hits the ground
B Find the maximum altitude achieved by the...
Homework Statement
Homework Equations
The Attempt at a Solution
a) (∂z/∂x)=-(∂f/∂x)*(∂z/∂f)
i used that (AxB)=-(BxA)
so i get
(∂z/∂x)=-[-(∂z/∂f)(∂f/∂x)]
=(∂z/∂x)
is this correct if not can someone give me hints pls
thanks
1. Hey all, For my calculus class we were giving the problem of solving for the optimization of a tin can using differential calculus. The problem was to find the minimum cost for any tin can of any height(as well as using the equation for the tin we had). The surface area of the cylinder was...
Homework Statement
A spherical balloon is being inflated at the rate of 10 cu in/sec. Find the rate of change of the area when the balloon has a radius of 6 in.
Homework Equations
V = \frac {4}{3} \pi r^{3} and A = 4 \pi r^{2}
The Attempt at a Solution
\frac {dV}{dt} =...
Homework Statement
the height of a certain hill is given by
h(x,y)=10(2xy-3x^2-4y^2-18x+28y+12)
wher y is the distance north and x is the distance east of south hadley
a)Where is the top of the hill
b) how high is the hill
c) how steep is the hill at a point 1 mile north...
Homework Statement
Find f'(x)= 16x - x-2 using first principles.
Homework Equations
x
http://img153.imageshack.us/img153/8403/597137697c1f605c7a43d34qz4.png
The Attempt at a Solution
I used dy/dx and got 2x-3 + 16 but I get something different when I use the formula I attempted...
Ok I am doing 1st year maths at uni and I am finding the differential calculus course really hard, i was hoping people here could just help me with the ideas.
firstly:
Level curves - I am having trouble drawing out the level curves for functions of two variables.
For f(x,y) = 2x + y - 5...
Any help is greatly appreciated! :smile:
Homework Statement
1.Differentiate.
http://www.webassign.net/www16/symImages/8/a/e5af282af9dd30006849e16c0b489b.gif
2.Find the derivative of the function.
http://www.webassign.net/www16/symImages/c/8/fd38a158e810bff80a28202fbceb37.gif...
I need a refresher on my multivariable differential calculus. Does anyone know of something brief with lots of exercises (maybe something geared towards physicists)?
Just a question regarding Differential Calculus Application. Can anyone help me to solve this using differential calculus. Here it is:
A farmer has enough money to build only 100 meters of fence. What are the dimensions of the field he can enclose the maximum area?
thanks in advanced:D
English translation of Fichtenholz, "Integral & differential calculus"
I've been recommended the book
"Integral & differential calculus, vol I,II,III by G M Fichtenholz.""
however I can only find a german version: "Differentialrechnung und Integralrechnung..."
Would anyone know if...
This is biological application question, so the limit can't be negative infinity, which is what I would use as my answer for a regular question.
The rate at which substance is eliminated from the body is proportional to the amount of substance present, and satisfies the differentail equation...
Frankly I don't know how to say this,I need some help In Differential Calculus.
I'm talking about Banach space and those stuff,If it's possible if someone know a link to some online textbook,i need something that goes deep into every theorm of this branch of math.
thx you already , you don't...
You are given the following information about the function f(x):
i) There is an x-value x* approximately equal to 0.8 such that f(x*)=0
ii) f(0.7) = C is negative
iii) m1 < f'(x) < m2 for 0.7 < x < 0.9 where m1 and m2 are positive constants
Apple the Mean Value Theorem to f(x) on the...
Please help.
For the function 1/(1-x) the Taylor polynomial of degree 3 about x=0 is:
p(x) = 1 + x + x^2 + x^3
a. Find an upper bound on l R(x) l if x = 0.5
b. Write down the first three non-zero terms of the Taylor series for
g(x) = 1/(4-x^2)
Is there an easier way to do b other...
hey i need help with 2 differential calculus problems, i missed the lecture so i am clueless as how to how to solve this.
i don't really want the answer, id rather someone show me the methods
anywhere, here goes:
2) A street light is at the top of a 16ft tall pole. A woman 6th tall walks...