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- (10point) Find points and on the parabola
so that the triangle
formed by the axis and the tangent lines at and is an equilateral
triangle.
(Solution)
Let be the coordinate of .
Since the derivative of ^ ^
is
the slope at is ^ .
But since the triangle is equilateral,
so the slope at is ^
Therefore we have
, and hence
Thus the point has coordinate ^ ^
and by symmetry, has coordinates
Answer :
- (10point) Let ^ ^ .
(a) For what value of is differentiable?
(b) Find a formula for the derivative (^) ′.
(solution)
If ^ ^ , ^ ^
(^) so ′ . If ,
(^) so
′^ ^ .
For ,
lim
→
^ ^ ^
lim
→
, and
lim
→
^ ^ ^
lim
→
(^)
.
Hence is differentiable at 0 and its derivative is 0. Hence, (a) is
differentiable for any , and (b) its derivative is
′^ ^
≥
(^) ≤
- (15point) Find an equation of the tangent line at which the tangent line to the
graph of the equation
(^)
(^) is parallel to the line (^) .
(solution)
First, let’s find ′ by implicit differentiation.
(^)
′ ^ ^ ′ implies that ′ ^
(^)
^
. We need to find a point at
which the tangent line is parallel to the line ^ , so let
(^)
^
(^) .
Hence
⇒ .
So ^ or ^ ^
Substituting for into the equation
results in
(^)
Hence or If , then , and if then .
However, at the point , ′
(^)
^
is undefined.
If ^ ^
, the equation
(^)
(^) can be rewritten as
(^) (^) (^) .
Substituting ^ ^
into this equation, we see ^
(^) . Hence, there is no
ln ^
⋅ln cos ^ ^ ^
Using L'Hospital's Rule,
lim
→
ln (^) lim
→
^
ln cos ^ ^ ^ ^
(^) lim
→
cos ^ ^ ^
(^) sin (^)
Hence, (^) lim
→
cos
By L'Hospital's Rule and the Fundamental Theorem of Calculus,
lim
→
cos (^) (^)
(^) lim
→
cos
7. (15point) Evaluate the limit
lim
→ ∞
cos
sin
(solution)
lim
→ ∞
^
cos
sin
^ cos
sin
^
^
tan
(^)
- (10point) Find the volume of the solid obtained by rotating the region bounded
by cos ,
,
and about .
(Solution)
cos
cos
cos
^
^
cos
^ cos
(^) (^) sin (^) sin
(^) (^)
