Fall, 2015, Problem Set 4 (Exam 1 Review)
1. Find the value of x so that the vectors h 4 , x + 1 i and h x, 3 i are perpendicular.
2. A woman exerts a horizontal force of 65 lb on a crate as she pushes it up a ramp that is 20 f t long and inclined at an angle of 20 ◦ above the horizontal. Find the work done on the box.
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3. Find the vector and parametric equations of the line passing through the points (7 , − 3) and parallel to the line x + 3 y = 3.
4. Consider the curve x = 3 + cos t , y = − 1 + sin t . Eliminate the parameter to find a Cartesian equation, and sketch the curve.
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5. Find the angles of the triangle with the given vertices.
A (3 , 0) , B (5 , 6) , C ( − 2 , 1)
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6. Find the vector and scalar projections of − 3 i + j onto 2 i + 5 j .
7. Find the distance from the point P (1 , 1) to the line y = − x + 4.
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8. Evaluate lim x →−∞
− x 3 + 2 x
8 + 4 x 2
2 − 4 x
.
− 5 x 3
9. Evaluate lim x →−∞
√
9 x 2 + 4 x
.
4 x + 1
10. Find lim x →
5 +
5 x − x 2
(5 − x ) 2
.
11. Evaluate lim x →−
3 x 2 + 3 x
.
| x + 3 |
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12. Find the horizontal and vertical asymptotes for f ( x ) =
(2 − x )(3 x + 1)
.
x 2 − 4
13. Fine lim t →
4 r ( t ) where r ( t ) = 2 t + 1 ,
√ t + 5 − 3 t − 4
.
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14. From the accompaning figure, state the numbers at which f is discontinuous.
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15. Given f ( x ) =
( x − 4 a ax 2 at x = 2.
if x < − 2 if x ≥ − 2
. Find the value of a which makes the function continuous
16. Which interval contains a solution to the equation x 3 + x = 3.
(a) [ − 1 , 0]
(b) [0 , 2]
(c) [0 , 1]
(d) [ − 2 , − 1]
(e) [2 , 4]
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17. Find the average rate of change of f ( t ) =
√
2 t + 3 from t = 1 to t = 3.
18. For f ( x ) = 2 x 2 − x + 1, find f ′ ( x ) using the definition of the derivative.
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19. For f ( x ) =
1
2 x + 1
, find f ′ ( x ) using the definition of the derivative.
20. If f (2) = 3 and f ′ (2) = − 7, find the equation of the tangent line to the graph of f ( x ) at x = 2.
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