Calculus: Analytic Geometry and Calculus, with Vectors

186 Functions, limits, derivatives 19 The vector formula r = (b + a cos 0)cos fi + (b + a cos O)sin Oj + a sin Ok of Problem 22 ...

3.7 Rates, velocities 187 22 Let a particle of mass m move in a vertical plane in such a way that its coordinates x, y are diffe ...

188 Functions, limits, derivatives Our knowledge of the natures of graphs of lines now shows that the graph of y = I(x) over the ...

3.8 Related rates 189 man is rising -* units per second and is therefore falling -- units per second, and our problem is solved. ...

190 Functions, limits, derivatives When two of the three numbers in this formula are known, we can calcu- late the remaining one ...

3.8 Related rates 191 to the spool atthe point T. Find the rate of increase of the distance from the axis of the spool to the en ...

192 Functions, limits, derivatives 10 Let 0 be the angle between the lines of Figure 3.893 that have lengths b and x. Show that ...

3.9 Increments and differentials 193 Even though we have written (1) in such a way that t does not appear, we rise to the occasi ...

194 Functions, limits, derivatives lies in the fact that small relative errors in the large weights can produce huge relative er ...

3.9 Increments and differentials 195 Such numbers dy and dx are called differentials, and some useful observa- tions can be made ...

196 Functions, limits, derivatives and then multiply by dx. A little experience with these things makes us realize that if y = s ...

3.9 Increments and differentials 197 If we measure the edges of a cube and decide that, subject to errors in measurement, each s ...

198 Functions, limits, derivatives 4 (1) Suppose that x and y are differentiable functions of t such that x2+y2 = 1. Show that d ...

3.9 Increments and differentials 199 is equivalent to the other two. Use these formulas to obtain (2) dT = .(:)-bigdL 2Ldg g g ( ...

200 Functions, limits, derivatives where g is the (scalar) acceleration of gravity, vo is the initial speed, and a is the angle ...

3.9 Increments and differentials 201' better. The specific heat o (sigma) of a substance at temperaturex is Q'(x), Where Q(x) is ...

4 Integrals 4.1 Indefinite integrals There are about as many different types of integrals in mathematics as there are elements i ...

4.1 Indefinite integrals 203 which (4.11) holds, we represent it by the ingenious symbol in the formula (4.111) F(x) = ff(x) dx ...

(^204) Integrals Theorem 4.13 If two functions y and F have the same derivative overan interval, then there is a constant c such ...

4.1 Indefinite integrals 205 The following little table gives two versions and most useful integration formulas. xn+i (4.171) / ...