Calculus derivative Definition and 19 Threads

I'm teaching Calc I. this semester and we're now covering the derivatives of power function and exponential functions as well as the basic rules, e.g. linearity and product rule. Some years back I ran across an exposition of umbral calculus in the appendix of a reference. I cannot help but...

We define differentiation as the limit of ##\frac{f(x+h)-f(x)}h## as ##h->0##. We find the instantaneous velocity at some time ##t_0## using differentiation and call it change at ##t_0##. We show tangent on the graph of the function at ##t_0##. But after taking h or time interval as zero to find...

I have attached an image of a function that I fit to a scatter plot, and I would like to know if there is a term for the point on the function at which the slope transitions from being less than -1 to greater than -1. I have highlighted this point approximately in yellow...

A plane with equation xa+yb+zc=1 (a,b,c>0) together with the positive coordinate planes forms a tetrahedron of volume V=16abc. Find the plane that minimizes V if the plane is constrained to pass through a point P=(9,6,3) . HELP! I have fount the partial derivatives for both of the given...

Homework Statement Simon suffers an injury at a campsite on the bank of a canal that is 6sqrt(2) km wide. if he is able to paddle his kayak at 5km/h and pedal a bicycle at 15km/h along the opposite bank, where should be land on the opposite bank to reach the hospital in minimum time? Homework...

Consider the following two calculations: (1) d(x\cos x)=(\cos x-x\sin x)dx (2) \frac{d}{dx}(x\cos x)=(\cos x-x\sin x) I would describe these both as differentiation. Is there a standard terminology that allows one to make the distinction between the two, if desired? The best I could come up...

So I started with limits and everything went well, and then into derivatives, and a problem started. First I know how to find the derivative of basic things using like ( limit definition of derivative) or (rules for derivatives) or ( product rule/ quotient rule/ chain rule)... My problem...

Homework Statement Let A be a set of critical points of the function f(x). Let B be a set of roots of the equation f''(x)=0. Let C be a set of points where f''(x) does not exist. It follows that B∪C=D is a set of potential inflection points of f(x). Q 1: Can there exist any inflection points...

Suppose that we have this metric and want to find null paths: ds^2=-dt^2+dx^2 We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$ dx=\pm dt \to x=\pm t Now switch to the more abstract and rigorous one-forms in differentiable manifolds. Here \mathrm{d}t (v)...

Suppose that we have this metric and want to find null paths: ds^2=-dt^2+dx^2 We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$ dx=\pm dt \to x=\pm t Now switch to the more abstract and rigorous one-forms in differentiable manifolds. Here \mathrm{d}t (v)...

Homework Statement The revenue function for a product is r = 8x where r is in dollars and x is the number of units sold. the demand function is q = -1/4p + 10000 where q units can be sold when selling price is p. what is dr/dp? Homework Equations r=pq The Attempt at a Solution I substituted...

Homework Statement A police car is parked 50 feet away from a wall. The police car siren spins at 30 revolutions per minute. What is the velocity the light moves through the wall when the beam forms angles of: a) α= 30°, b) α=60°, and c) α=70°? This is the diagram...

The function f: R → R is: f(x) = (tan x) / (1 + ³√x) ; for x ≥ 0, sin x ; for (-π/2) ≤ x < 0, x + (π/2) ; for x < -π/2 _ For the interval (0,∞), we are interested in f such that f(x) = (tan x) / (1 + ³√x) ; for x ≥ 0 f(x) = tan x / (1 + x¹ʹ³)            (1 + x¹ʹ³)•sec²x −...

Question: Show that the derivative of f(x) = (x-a)m (x-b)n vanishes at some point between a and b if m and n are positive integers. My attempt: f(x) = (x-a)m (x-b)n f '(x) = m(x-a)m-1 (x-b)n + n(x-a)m (x-b)n-1 f '(x) = [(x-a)n-1 (x-b)n-1 ] [(m)(x-b) +(n)(x-a)] And this is as far as I got.

Homework Statement Calculus Derivative help? when x = 0, f=2, f '=1, g=5, g '= -4 when x = 1, f=3, f '=2, g=3, g '= -3 when x = 2, f=5, f '=3, g=1, g '= -2 when x = 3, f=10, f '=4, g=0, g '= -1 based on the above table, if a = f +2g, then a'(3) =? please show steps.thanks Homework...

Homework Statement derivative of (e^x-e^-x)/(e^x+e^-x) Homework Equations The Attempt at a Solution This answer was marked 2/4 on the test, can anyone help me get the right answer please ?

Homework Statement Find the derivative of g(x)=x * sqrt(4-x)Homework Equations The Attempt at a Solution I know I use the product rule, but I am not sure how to derive the sqrt(4-x) portion.

Hey... i was hopeing somebody can help me with a homework question... its about vector calc... taking the deriviative. d/dx[r(t) dot r(t)] = r'(t) dot r(t) + r(t) dot r'(t) = 2r'(t) dot r(t) I know it sounds sillly, but i was just wondering how on Earth they got 2r'(t) dot r(t) ?

Hey, Anyone out there able to help me out? I'm trying to find y'' of the equation y=xtanx. I found y' to equal 1(sec²x) but I don't know what to do after that. I know the final answer should be 2cosx + 2xsinx/ cos³x after being simplified and stuff, but I am clueless as to how to get there...