Exercise 2.1
(i) The curve y = 1/(1 + x2) is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point .
(ii) The curve y = x/(1 + x2) is called a serpentine. Find an equation of the tangent line to this curve at the point (3,0.3).
Exercise 2.2 [Ste10, p. 190] If f is a differentiable function, find an expression for the derivative of each of the following functions.
(i) y = x2f(x)
Exercise 2.3 [Ste10, p. 191]
(a) If g is differentiable, the Reciprocal Rule says that
Use the Quotient Rule to prove the Reciprocal Rule.
(b) Use the Reciprocal Rule to verify that the Power Rule is valid for negative integers, that is,
Exercise 2.4 [Ste10, p. 197] Calculate the first and second derivatives of the following functions.
√
(i) f(x) = xsinx
(viii) f(x) = x2 sinxtanx
Exercise 2.5 [Ste10, p. 197]
(a) Use the Quotient Rule to differentiate the function
(b) Simplify the expression for f(x) by writing it in terms of sinx and cosx, and then find f0(x).
(c) Show that your answers to parts (a) and (b) are equivalent.
Exercise 2.6 [Ste10, p. 198] Find the limit (use whatever method you like, but show the details of your work)
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Exercise 2.7 [Ste10, p. 198]
(a) Evaluate .
(b) Evaluate .
(c) Illustrate parts (a) and (b) by graphing y = sin(1/x).
Exercise 2.8 [Ste10, p. 198] Find constants A and B such that the function y = Asinx + B cosx satisfies the differential equation y00 + y0 − 2y = sinx.
Exercise 2.9 Given function f satisfying |f(x)| ≤ x2, calculate f0(0).
