Vv156 Homework 3 -Solved

(i)   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).

(ii) 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.

(i)   The derivative of an even function is an odd function.

(ii)  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 dx2

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

(i)   (cardioid) .

(ii)    (astroid) x2/3 + y2/3 = 4 at (−3 3,1).

(iii)   (lemniscate) 2(x2 + y2)2 = 25(x2 − y2) at (3,1).

(iv)  (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 

(i) Show that
 
(sinhx) = coshx
d

(coshx) = sinhx
     d                                   2

(tanhx) = sech x
                              dx                                                                 dx                                                                  dx

                              (cschx) =        cschxcothx

dx

(ii) (3pts) and show that1
       (sechx) =        sechxtanhx

dx
           (cothx) =         csch x

dx
  d                       d       d              2 −−−

                                                                                                                                                                          d             −1                          1

                                                                                                                                                                             (tanh       x) =

  dx            1 − x2 d  −1                   1

                                                                                                                                                                              (coth       x) =

                                                                                                                                                                         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.

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

(ii)  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

(i)  Show that f is continuous at 0.

(ii) 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