By Vincent Orange
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Additional resources for Coningham - A Biography of Air Marshall Sir Arthur Coningham
Dx v v2 (13) (v) Derivative of a function of a function: Given the function y[u(x)], dy du dy = · . dx du dx (14) Equation (14) leads immediately to the relations dy dy dx = du if =0 dx dx du du and dx dy 1 if = = 0. dx dx dy dy (vi) The power formula: du d n u = nun−1 dx dx (15) for the function u(x) raised to any power. The derivative of the logarithm was already discussed in Chapter 1, while the derivatives of the various trigonometric functions can be developed from their deﬁnitions [see, for example, Eqs.
Given x = t 2 + 2t − 4 and y = t 2 − t + 2, evaluate d2 y/dx 2 . Ans. cf. 5. 8. If y = A cos kx + B sin kx, where A, B and k are constants, ﬁnd the expression for d2 y/dx 2 . d2 y Ans. = −k 2 y dx 2 9. Verify the series for cos ϕ and sin ϕ given by Eqs. (1-34) and (1-35), respectively. 10. Verify Eq. (53) and derive the other three Maxwell relations, namely, ∂S ∂V 11. = T ∂P ∂T , V ∂T ∂P = S ∂V ∂S and P ∂S ∂P =− T ∂V ∂T . P Find the ﬁrst partial derivatives of the function z = 4x 2 y − y 2 + 3x − 1.
19. Repeat problem 18 for the function 3i/(i − 3). 18. 20. Ans. 1 Ans. 3/2 Given the deﬁnitions cos ϕ = (eiϕ + e−iϕ )/2 and, sin ϕ = (eiϕ − e−iϕ )/2i, show that cos(ϕ + γ ) = cos ϕ cos γ − sin ϕ sin γ and therefore, cos[(π/2) − ϕ] = sin ϕ. 21. Given the deﬁnitions of the functions sinh and cosh, prove Eq. (48). √ 22. Show that sinh−1 x = ln(x + x 2 + 1), x > 0. ∗ Daniel Gabriel Fahrenheit, German physicist (1686–1736). † Anders Celsius, Swedish astronomer and physicist (1701–1744). 1 DEFINITION OF A LIMIT Given a function y = f (x) and a constant a: If there is a number, say γ , such that the value of f (x) is as close to γ as desired, where x is different from a, then the limit of f (x) as x approaches a is equal to γ .
Coningham - A Biography of Air Marshall Sir Arthur Coningham by Vincent Orange