Calculus Answers

f(x,y)=arctan(2x/(x^2+y^2))

Find the area of the triangle formed from the coordinate axes and the tangent line to the curve y = 5x^(−1) −x/5 at the point (5,0).

An equilateral triangle made of metallic sheet is expanding because it is being heated. Its area A is given by

A =(√3x^2)/4

square centimeters, where x is the length of one side in centimeters. Find the instantaneous rate of change in A with respect to x at the instant when x = 10 centimeters.

If g(x)=cos 2x. Find g(\pi /4), g(\pi /2), g(\pi -x), g(\pi +x), g(x-\pi /2)

Write and draw the graph on matlab or octave online of a function which is

(a) Continuous on all points except at x = 1.

(b) Differentiable on all points except at x = 1.

(c) Non-differentiable at five points x = 1, x = 2, x = 3, x = 4 and x = 5.

Find the domain and range of the following function given by

𝑓(𝑥)=√(3𝑥−5)(𝑥+4)/𝑥3−16𝑥

The force F (in pounds) acting at an angle θ with the horizontal that is needed to drag a crate weighing W pounds along a horizontal surface at a constant velocity is given by

F = μW/(cosθ +μsinθ)

where μ is a constant called the coefficient of sliding friction between the crate and the surface (see the accompanying figure). Suppose that the crate weighs 150 lb and that μ = 0.3.

(a) Find dF /dθ when θ =30°. Express the answer in units of pounds/degree.

(b) Find dF /dt when θ =30° if θ is decreasing at the rate of 0.5°/s at this instant.

In the temperature range between 0°C and 700°C the resistance R [in ohms] of a certain platinum resistance thermometer is given by

R = 10 + 0.04124T − 1.779 × 10^(−5)T^2

where T is the temperature in degrees Celsius. Where in the interval from 0°C to 700°C is the resistance of the thermometer most sensitive and least sensitive to temperature changes? [Hint: Consider the size of dR/dT in the interval 0 ≤ T ≤ 700.].

Question 03

Write (and draw the graph of) a function which is

(a) Continuous on all points except at x = 1.

(b) Differentiable on all points except at x = 1.

(c) Non-differentiable at five points x = 1, x = 2, x = 3, x = 4 and x = 5.