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Math Calculus UD CALC (241 ONLY) W/1 TERM ACCESS >IB

Establish the following rules for working with differentials (where c denotes a constant and u and v are functions of x). (a) d c = 0 (b) d ( c u ) = c d u (c) d ( u + v ) = d u + d v (d) d ( u v ) = u d v + v d u (e) ( u v ) = v d u − u d v v 2 (f) d ( x n ) = n x n − 1 d x

Establish the following rules for working with differentials (where c denotes a constant and u and v are functions of x). (a) d c = 0 (b) d ( c u ) = c d u (c) d ( u + v ) = d u + d v (d) d ( u v ) = u d v + v d u (e) ( u v ) = v d u − u d v v 2 (f) d ( x n ) = n x n − 1 d x

Solution Summary: The author explains how to establish the given rules using differentials.

UD CALC (241 ONLY) W/1 TERM ACCESS >IB

UD CALC (241 ONLY) W/1 TERM ACCESS >IB

8th Edition

ISBN: 9781337051545

Author: Stewart

Publisher: CENGAGE C

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Textbook Question

Chapter 2.9, Problem 39E

Establish the following rules for working with differentials (where c denotes a constant and u and v are functions of x).

(a) d c = 0

(b) d ( c u ) = c d u

(c) d ( u + v ) = d u + d v

(d) d ( u v ) = u d v + v d u

(e) ( u v ) = v d u − u d v v 2

(f) d ( x n ) = n x n − 1 d x

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Chapter 2.9, Problem 38E

Chapter 2.9, Problem 40E

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Chapter 2 Solutions

UD CALC (241 ONLY) W/1 TERM ACCESS >IB

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Find the...Ch. 2.7 - Prob. 16E Ch. 2.7 - The mass of the part of a metal rod that lies...Ch. 2.7 - If a tank holds 5000 gallons of water, which...Ch. 2.7 - The quantity of charge Q in coulombs C that has...Ch. 2.7 - Prob. 20E Ch. 2.7 - Prob. 21E Ch. 2.7 - Some of the highest tides in the world occur in...Ch. 2.7 - Boyles Law states that when a sample of gas is...Ch. 2.7 - Prob. 24E Ch. 2.7 - The table gives the population of the world Pt, in...Ch. 2.7 - Prob. 26E Ch. 2.7 - Prob. 27E Ch. 2.7 - Prob. 28E Ch. 2.7 - Prob. 29E Ch. 2.7 - The cost function for a certain commodity is...Ch. 2.7 - If px is the total value of the production when...Ch. 2.7 - Prob. 32E Ch. 2.7 - The gas law for an ideal gas at absolute...Ch. 2.7 - Invasive species often display a wave of advance...Ch. 2.7 - In the study of ecosystems, predator-prey models...Ch. 2.7 - In a fish farm, a population of fish is introduced...Ch. 2.8 - Prob. 1E Ch. 2.8 - a If A is the area of a circle with radius r and...Ch. 2.8 - Each side of a square is increasing at a rate of 6...Ch. 2.8 - Prob. 4E Ch. 2.8 - A cylindrical tank with radius 5 m is being filled...Ch. 2.8 - Prob. 6E Ch. 2.8 - The radius of a spherical ball is increasing at a...Ch. 2.8 - The area of a triangle with sides of lengths a and...Ch. 2.8 - Suppose y=2x+1, where x and y are functions of t....Ch. 2.8 - Prob. 10E Ch. 2.8 - Prob. 11E Ch. 2.8 - A particle is moving along a hyperbola xy=8. As it...Ch. 2.8 - 13-16 a What quantities are given in the problem?...Ch. 2.8 - 13-16 a What quantities are given in the problem?...Ch. 2.8 - 13-16 a What quantities are given in the problem?...Ch. 2.8 - Prob. 16E Ch. 2.8 - Two cars start moving from the same point. One...Ch. 2.8 - Prob. 18E Ch. 2.8 - A man starts walking north at 4 ft/s from a point...Ch. 2.8 - A baseball diamond is a square with side 90 ft. A...Ch. 2.8 - The altitude of a triangle is increasing at a rate...Ch. 2.8 - A boat is pulled into a dock by a rope attached to...Ch. 2.8 - At noon, ship A is 100 km west of ship B. Ship A...Ch. 2.8 - Prob. 24E Ch. 2.8 - Water is leaking out of an inverted conical tank...Ch. 2.8 - A trough is 10 ft long and its ends have the shape...Ch. 2.8 - A water trough is 10 m long and a cross-section...Ch. 2.8 - A swimming pool is 20 ft wide, 40 ft long, 3 ft...Ch. 2.8 - Gravel is being dumped from a conveyor belt at a...Ch. 2.8 - A kite 100 ft above the ground moves horizontally...Ch. 2.8 - Prob. 31E Ch. 2.8 - Prob. 32E Ch. 2.8 - The top of a ladder slides down a vertical wall at...Ch. 2.8 - According to the model we used to solve Example 2,...Ch. 2.8 - If the minute hand of a clock has length r in...Ch. 2.8 - Prob. 36E Ch. 2.8 - Prob. 37E Ch. 2.8 - When air expands adiabatically without gaining or...Ch. 2.8 - Prob. 39E Ch. 2.8 - Brain weight B as a function of body weight W in...Ch. 2.8 - Prob. 41E Ch. 2.8 - Two carts, A and B, are connected by a rope 39 ft...Ch. 2.8 - Prob. 43E Ch. 2.8 - Prob. 44E Ch. 2.8 - A plane flies horizontally at an altitude of 5 km...Ch. 2.8 - A Ferris wheel with a radius of 10 m is rotating...Ch. 2.8 - A plane flying with a constant speed of 300 km/h...Ch. 2.8 - Two people start from the same point. One walks...Ch. 2.8 - A runner sprints around a circular track of radius...Ch. 2.8 - The minute hand on a watch is 8 mm long and the...Ch. 2.9 - 14 Find the linearization L(x) of the function at...Ch. 2.9 - Prob. 2E Ch. 2.9 - 14 Find the linearization L(x) of the function at...Ch. 2.9 - Prob. 4E Ch. 2.9 - Prob. 5E Ch. 2.9 - Find the linear approximation of the function...Ch. 2.9 - Prob. 7E Ch. 2.9 - Prob. 8E Ch. 2.9 - 7-10 Verify the given linear approximation at a=0....Ch. 2.9 - Prob. 10E Ch. 2.9 - 11-14 Find the differential dy of each function. a...Ch. 2.9 - 11-14 Find the differential dy of each function. a...Ch. 2.9 - 11-14 Find the differential dy of each function. a...Ch. 2.9 - Prob. 14E Ch. 2.9 - 15-18 a Find the differential dy and b evaluate dy...Ch. 2.9 - Prob. 16E Ch. 2.9 - Prob. 17E Ch. 2.9 - Prob. 18E Ch. 2.9 - 19-22 Compute y and dy for the given values of x...Ch. 2.9 - Prob. 20E Ch. 2.9 - 19-22 Compute y and dy for the given values of x...Ch. 2.9 - Prob. 22E Ch. 2.9 - 23-28 Use a linear approximation or differentials...Ch. 2.9 - Prob. 24E Ch. 2.9 - Prob. 25E Ch. 2.9 - 23-28 Use a linear approximation or differentials...Ch. 2.9 - Prob. 27E Ch. 2.9 - Prob. 28E Ch. 2.9 - 29-30 Explain, in terms of linear approximations...Ch. 2.9 - Prob. 30E Ch. 2.9 - The edge of a cube was found to be 30 cm with a...Ch. 2.9 - The radius of a circular disk is given as 24 cm...Ch. 2.9 - The circumference of a sphere was measured to be...Ch. 2.9 - Use differentials to estimate the amount of paint...Ch. 2.9 - a Use differentials to find a formula for the...Ch. 2.9 - One side of a right triangle is known to be 20 cm...Ch. 2.9 - Prob. 37E Ch. 2.9 - When blood flows along a blood vessel, the flux F...Ch. 2.9 - Establish the following rules for working with...Ch. 2.9 - Prob. 40E Ch. 2.9 - Suppose that the only information we have about a...Ch. 2.9 - Suppose that we dont have a formula for g(x) but...Ch. 2.R - Write an expression for the slope of the tangent...Ch. 2.R - Prob. 2CC Ch. 2.R - Prob. 3CC Ch. 2.R - Define the derivative fa. Discuss two ways of...Ch. 2.R - Prob. 5CC Ch. 2.R - Prob. 6CC Ch. 2.R - Prob. 7CC Ch. 2.R - Prob. 8CC Ch. 2.R - Prob. 9CC Ch. 2.R - Prob. 10CC Ch. 2.R - Give several examples of how the derivative can be...Ch. 2.R - Prob. 12CC Ch. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 2TFQ Ch. 2.R - Prob. 3TFQ Ch. 2.R - Prob. 4TFQ Ch. 2.R - Prob. 5TFQ Ch. 2.R - Determine whether the statement is true or false....Ch. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 8TFQ Ch. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 10TFQ Ch. 2.R - Prob. 11TFQ Ch. 2.R - Prob. 12TFQ Ch. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 14TFQ Ch. 2.R - Prob. 15TFQ Ch. 2.R - The displacement in meters of an object moving in...Ch. 2.R - The graph of f is shown. State, with reasons, the...Ch. 2.R - Prob. 3E Ch. 2.R - Prob. 4E Ch. 2.R - The figure shows the graph of f,f, and f".Identify...Ch. 2.R - Prob. 6E Ch. 2.R - The total cost of repaying a student loan at an...Ch. 2.R - Prob. 8E Ch. 2.R - Prob. 9E Ch. 2.R - Prob. 10E Ch. 2.R - Prob. 11E Ch. 2.R - Prob. 12E Ch. 2.R - Prob. 13E Ch. 2.R - Prob. 14E Ch. 2.R - 1340 Calculate y. y=x2x+2x Ch. 2.R - Prob. 16E Ch. 2.R - Prob. 17E Ch. 2.R - Prob. 18E Ch. 2.R - Prob. 19E Ch. 2.R - Prob. 20E Ch. 2.R - Prob. 21E Ch. 2.R - Prob. 22E Ch. 2.R - Prob. 23E Ch. 2.R - Prob. 24E Ch. 2.R - Prob. 25E Ch. 2.R - Prob. 26E Ch. 2.R - Prob. 27E Ch. 2.R - Prob. 28E Ch. 2.R - 1340 Calculate y. sin(xy)=x2y Ch. 2.R - Prob. 30E Ch. 2.R - Prob. 31E Ch. 2.R - Prob. 32E Ch. 2.R - Prob. 33E Ch. 2.R - Prob. 34E Ch. 2.R - Prob. 35E Ch. 2.R - Prob. 36E Ch. 2.R - Prob. 37E Ch. 2.R - Prob. 38E Ch. 2.R - Prob. 39E Ch. 2.R - Prob. 40E Ch. 2.R - If f(t)=4t+1, find f"(2).Ch. 2.R - Prob. 42E Ch. 2.R - Find y" if x6+y6=1.Ch. 2.R - Prob. 44E Ch. 2.R - Prob. 45E Ch. 2.R - Prob. 46E Ch. 2.R - Prob. 47E Ch. 2.R - Prob. 48E Ch. 2.R - Prob. 49E Ch. 2.R - Prob. 50E Ch. 2.R - Prob. 51E Ch. 2.R - Prob. 52E Ch. 2.R - Prob. 53E Ch. 2.R - Prob. 54E Ch. 2.R - Prob. 55E Ch. 2.R - How many tangent fines to the curve y=x/(x+1) pass...Ch. 2.R - If f(x)=(xa)(xb)(xc), show that...Ch. 2.R - Prob. 58E Ch. 2.R - Prob. 59E Ch. 2.R - If f and g are the functions whose graphs are...Ch. 2.R - Prob. 61E Ch. 2.R - Prob. 62E Ch. 2.R - Prob. 63E Ch. 2.R - Prob. 64E Ch. 2.R - Prob. 65E Ch. 2.R - Prob. 66E Ch. 2.R - Prob. 67E Ch. 2.R - Prob. 68E Ch. 2.R - Prob. 69E Ch. 2.R - 6971 Find h in terms of f.and g. h(x)=f(x)g(x)Ch. 2.R - Prob. 71E Ch. 2.R - Prob. 72E Ch. 2.R - A particle moves on a vertical line so that its...Ch. 2.R - The volume of a right circular cone is V=13r2h,...Ch. 2.R - The mass of part of a wire is x(1+x) kilograms,...Ch. 2.R - The cost, in dollars, of producing x units of a...Ch. 2.R - Prob. 77E Ch. 2.R - Prob. 78E Ch. 2.R - Prob. 79E Ch. 2.R - A waterskier skis over the ramp shown in the...Ch. 2.R - Prob. 81E Ch. 2.R - Prob. 82E Ch. 2.R - a Find the linearization of f(x)=1+3x3 at a=0....Ch. 2.R - Prob. 84E Ch. 2.R - A window has the shape of a square surmounted by a...Ch. 2.R - 8688 Express the limit as a derivative and...Ch. 2.R - Prob. 87E Ch. 2.R - Prob. 88E Ch. 2.R - Prob. 89E Ch. 2.R - Prob. 90E Ch. 2.R - Prob. 91E Ch. 2.R - Show that the length of the portion of any tangent...Ch. 2.P - Find points P and Q on the parabola y=1x2 so that...Ch. 2.P - Prob. 2P Ch. 2.P - Show that the tangent lines to the parabola...Ch. 2.P - Prob. 4P Ch. 2.P - Prob. 5P Ch. 2.P - Prob. 6P Ch. 2.P - Prob. 7P Ch. 2.P - Prob. 8P Ch. 2.P - Prob. 9P Ch. 2.P - Prob. 10P Ch. 2.P - How many lines are tangent to both of the circles...Ch. 2.P - If f(x)=x46+x45+21+x, calculate f(46)(3). Express...Ch. 2.P - The figure shows a rotating wheel with radius 40...Ch. 2.P - Tangent lines T1 and T2 are drawn at two points P1...Ch. 2.P - Let T and N be the tangent and normal lines to the...Ch. 2.P - Prob. 16P Ch. 2.P - Prob. 17P Ch. 2.P - Let P(x1,y1) be a point on the parabola y2=4px...Ch. 2.P - Suppose that we replace the parabolic mirror of...Ch. 2.P - Prob. 20P Ch. 2.P - Prob. 21P Ch. 2.P - Given an ellipse x2/a2+y2/b2=1, where ab, find the...Ch. 2.P - Prob. 23P Ch. 2.P - Prob. 24P Ch. 2.P - A lattice point in the plane is a point with...Ch. 2.P - Prob. 26P Ch. 2.P - Prob. 27P Ch. 2.P - Prob. 28P

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