Find by differentiation the slopes of the tangents to the following curves at the points indicated. Verify each result by drawing the curve and its tangent.
1. ,
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where .
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Ans. 4.
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2.
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where .
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-6.
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3. ,
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where .
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-3.
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4. ,
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where .
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.
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5. ,
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where .
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1.
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6. ,
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where .
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.
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7. ,
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where .
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4.
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8. ,
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where .
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0.
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9. ,
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where .
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6.
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10. Find the slope of the tangent to the curve
, (a) at the point where
; (b) at the point where
.
11. (a) Find the slopes of the tangents to the two curves
and
at their points of intersection. (b) At what angle do they intersect?
Ans. (a)
; (b)
.
12. The curves on a railway track are often made parabolic in form. Suppose that a track has the form of the parabola
(last figure, § 32), the directions
and
being east and north respectively, and the unit of measurement 1 mile. If the train is going east when passing through
, in what direction will it be going
(a) when mi. east of ?
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Ans. Northeast.
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(b) when mi. west of ?
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Southeast.
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(c) when mi. east of ?
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N. 30°E.
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(d) when mi. north of ?
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E. 30°S., or E. 30°N.
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13. A street-car track has the form of the cubical parabola
. Assume the same directions and unit as in the last example. If a car is going west when passing through
, in what direction will it be going
(a) when mi. east of ?
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Ans. Southwest.
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(b) when mi. west of ?
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Southwest.
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(c) when mi. north of ?
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S. 27° 43' W.
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(d) when 2 mi. south of ?
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(e) when equidistant from and ?
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