[SOLVED] Vv156 Honors Calculus II Assignment 2

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Exercise 2.1 [Ste10, p. 189]

• 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 .
•  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]

• If g is differentiable, the Reciprocal Rule says that

Use the Quotient Rule to prove the Reciprocal Rule.

• 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]

• Use the Quotient Rule to differentiate the function
• Simplify the expression for f(x) by writing it in terms of sinx and cosx, and then find f⁰(x).
• 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]

• Evaluate.
• Evaluate.
• 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).