Description
Note: Unless specified otherwise, you must show the details of your work via logical reasoning for each exercise. Simply writing a final result (whether correct or not) will receive 0 point.
Exercise 2.1 [Ste10, p. 189]
(i) (2 points) 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 (−1, 12 ).
(ii) (2 points) 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). (4 points)
Exercise 2.2 [Ste10, p. 190] If f is a differentiable function, find an expression for the derivative of each of the following functions.
2 f(x) x2 1+xf(x) (i)y=xf(x) (ii)y= x2 (iii)y=f(x) (iv)y= √x
(4 points)
Exercise 2.3 [Ste10, p. 191]
(a) If g is differentiable, the Reciprocal Rule says that d 1
dx g(x) Use the Quotient Rule to prove the Reciprocal Rule.
g′(x) = −[g(x)]2
(b) Use the Reciprocal Rule to verify that the Power Rule is valid for negative integers, that is,
d (x−n) = −nx−n−1 dx
(4 points)
Exercise 2.4 [Ste10, p. 197] Calculate the first and second derivatives of the following functions.
(i) f (x) = √x sin x (ii) f (x) = sin x + 1 cot x (iii) f (x) = 2 sec x − csc x (iv) f (x) = x
2 2−tanx
(v)f(x)= secx (vi)f(x)=xsinx (vii)f(x)=1−secx (viii)f(x)=x2sinxtanx 1+secx 1+x tanx
(8 points)
Exercise 2.5 [Ste10, p. 197]
(a) Use the Quotient Rule to differentiate the function
f(x)= tanx−1 sec x
(b) Simplify the expression for f(x) by writing it in terms of sinx and cosx, and then find f′(x). (c) Show that your answers to parts (a) and (b) are equivalent.
(3 points)
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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(i) lim sin 3x x→0 x
(v) lim sin3x x→0 5×3 −4x
(ix) lim 1−tanx x→π/4 sinx−cosx
(10 points)
Exercise 2.7 [Ste10, p. 198]
(a) Evaluate lim xsin1. x→∞ x
(b) Evaluate lim x sin 1 . x→0 x
(ii) lim sin 4x x→0 sin6x
(vi) lim sin3xsin5x x→0 x2
(x) lim sin(x−1) x→1 x2 +x−2
(iii) lim tan 6t t→0 sin2t
(vii) lim sinθ θ→0 θ+tanθ
(iv) lim cos θ θ→0 sinθ
(viii) lim 2x x→0 x+sinx
(c) Illustrate parts (a) and (b) by graphing y = sin(1/x). (3 points)
Exercise 2.8 [Ste10, p. 198] Find constants A and B such that the function y = A sin x + B cos x satisfies the differential equation y′′ + y′ − 2y = sin x.
(2 points)
Exercise 2.9 Given function f satisfying |f(x)| ≤ x2, calculate f′(0). (2 points)
References
[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on pages 1, 2).
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