(b) An astroid has an equation of the form x2/3+ y2/3= a2/3, where a is a positive constant

(1)

93 ç,ç‚B 1 ‚255æ 1. (25 points)

(a) Find d dx

√3

sec x + tan x and d dx

√ 1

1 + x²(x +√

1 + x²).

(b) An astroid has an equation of the form x²^/3+ y²^/3= a²^/3, where a is a positive constant. Find dy

dx and show that the length of the portion of any tangent line to the astroid cut off by the coordinate axes is constant.

Solution. i. x²^/3+ y²^/3= a²^/3⇒ 2

3x⁻¹^/3+2

3y⁻¹^/3dy dx = 0.

⇒ dy

dx = −y¹^/3

x¹^/3 = −y x

¹/3

.

ii. Let (b, c) be on the curve, that is, b²^/3+ c²^/3 = a²^/3. At (b, c) the slope of the tangent line is −c

b

¹/3

and an equation of the tangent line is

y − c = −c b

¹^/3

(x − b) or y = −c b

¹^/3

x+ (c + b²^/3c¹^/3) Setting y = 0, we find that the x-intercept is

b¹^/3c²^/3+ b = b¹^/3(c²^/3+ b²^/3).

Setting x = 0, we find that the y-intercept is c+ b²^/3c¹^/3= c¹^/3(c²^/3+ b²^/3).

So the length of the tangent line between these two points is q

b¹^/3(c²^/3+ b²^/3)²

+c¹^/3(c²^/3+ b²^/3)²

= q

b²^/3 a²^/3²

+ c²^/3 a²^/3²

= q

b²^/3+ c²^/3 a⁴^/3=p

a²^/3a⁴^/3=√

a²= a = constant 2. (20 points) Evaluate the following limits.

(a) lim

n→∞

√1 n

n

X

k=1

√1 k, (b) lim

x→∞x Z ⁴x

2x

1

√3

t⁶+ 100dt.

1

(2)

3. (20 points) Let

F(x) = Z ⁻¹

x³

(x³− t)f (√³

t)dt, x <0,

where f (x) is a continuous function. Suppose F (x) is an antiderivative of x⁴, find F (x) and f (x).

4. (15 points) Among the tangent lines of the graph y = x²−³2, find the one nearest to the origin.

(ÊÇ$ y = x²−³2 Fí~(52, v|ø‘(DŸõí×|¡ ) 5. (20 points) Graph the function (x + 1)³

(x − 1)² that reveal all the important as- pects of the curve such as symmetry, the intervals of increase and decrease, extreme values, intervals of concavity, inflection points, and asymptotes.

2