Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

62 Differentiation of Functions [Ch. 2
These segments are expressed by the following formulas:
621. What angles cp are formed with the x-axis by the tangents
to the curve y = x x 2
at points with abscissas:
a) x = 0; b) x=l/2; c) x=l?
Solution. We have y'^\ 2x. Whence
a) tan cp = l, =0;
c) tan q> = 1, q> = 135 (Fig. 15).
622. At what angles do the sine
and y= s'm2x inter-
2 N sect the axis of abscissas at the
origin?
Fig. 15 623. At what angle does the tangent
curve y = ianx intersect the
axis of abscissas at the origin?
624. At what angle does the curve y = e*' tx
intersect the
straight line x = 2?
625. Find the points at which the tangents
y =z 3* 4
-f. 4x* 12x* + 20 are parallel to the jc-axis.
626. At what point is the tangent to the parabola
to the curve
parallel to the straight line 5x + y 3 = 0?
627. Find the equation of the parabola y~x*-}-bx-\-c that is
tangent to the straight line x = y at the point (1,1).
628. Determine the slope of the tangent to the curve x*+y*
xy7 = Q at the point (1,2).
629. At what point of the curve y 2 = 2x* is the tangent perpendicular
to the straight line 4x3y + 2 = 0?
630. Write the equation of the tangent and the normal to the
parabola
,/y
= K x
at the point with abscissa x = 4.
f
Solution. We have y 7=; whence the slope of the tangent is
2
1
V x
k = [y']x= = i -T- Since the point of tangency has coordinates * = 4, y = 2, It
follows that the equation of the tangent is #2 = 1/4 (* 4) or x 4# + 4 = 0.
Since the slope of the normal must be perpendicular,
*, = -4;
whence the equation of the normal: t/2 = 4 (x 4) or 4x + y 18 0.