[SOLVED] Vv156 Honors Calculus II Assignment 3

Description

Exercise 3.1 [Ste10, p. 205]

• The curve y = |x|/ 2 − x² is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1,1).
•  Illustrate part (i) by sketch the curve and the tangent line on the same coordinate system.

Exercise 3.2 [Ste10, p. 208] Use the Chain Rule to prove the following.

• The derivative of an even function is an odd function.
• The derivative of an odd function is an even function.

Exercise 3.3 [Ste10, p. 208] If y = f(u) and u = g(x), where f and g are twice differentiable functions, show that

d

dx²            du²        dx             du dx

Exercise 3.4 [Ste10, p. 215] Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

(cardioid).

√

• ) (astroid) x^2/3 + y^2/3 = 4 at (−3 3,1).
• ( (lemniscate) 2(x² + y²)² = 25(x² − y²) at (3,1).
• ) (devil’s curve) y²(y² − 4) = x²(x² − 5) at (0,−2).

Exercise 3.5 [Ste10, Sec. 3.11] Given the following hyperbolic functions defined as

coth

dx                                                                 dx                                                                  dx

d                       d       d              ₂ −−−

d             −1                          1

(tanh       x) =

dx            1 − x² d  −1                   1

dx                            1 − x²

¹Notice that the formulas for the derivatives of tanh⁻¹x and coth⁻¹x appear to be identical. But the domains of these functions have no numbers in common: tanh⁻¹x is defined for |x| < 1, whereas coth⁻¹x is defined for |x| > 1.

Page 1 of

Exercise 3.6 [Ste10, p. 223] Find the derivative of the following functions (i) (1pt) y = (sinx)^lnx

(ii) (1pt) y = (tanx)^1/x

Exercise 3.7 [Ste10, p. 272] If

Show that.

Exercise 3.8 [Ste10, p. 282] If f has a local minimum value at c, show that the function g(x) = −f(x) has a local maximum value at c.

Exercise 3.9 [Ste10, p. 289] Suppose f is an odd function and is differentiable everywhere. Show that for every positive number b, there exists a number c ∈ (−b,b) such that f^′(c) = f(b)/b.

Exercise 3.10 [Ste10, p. 300] Show that the inflection points of the curve y = xsinx lie on the curve y^[1](x²+4) = 4x².

Exercise 3.11 [Ste10, p. 309] Evaluate

Exercise 3.12 [Ste10, p. 309] Let

• Show that f is continuous at 0.
• Calculate f^′(0).

Exercise 3.13 [Ste10, p. 309] Show that the shortest distance from the point (x₁,y₁) to the straight line Ax + By + C = 0