Question

1) Sketch the graph?=? ,?=? +3,and include orientation.

2) Sketch the graph ? = sin ? , ? = sin2 ? + 3 and include orientation.

3) Remove the parameter for ? = ? − 3, ? = ?2 + 3? − 2 and write the corresponding

rectangular equation.

4) Remove the parameter for ? = 2 + 3 sin ? , ? = −1 + 3 cos ? and write the corresponding rectangular equation.

5) Create a parameterization for ? = ?2 whose orientation is right to left.

For problem 6-10 use the parametric equations ? = ?2 + 5,? = ?2 − 3? − 1 .

6) Find ??/??

7) Find ?2?/??2

8) Find the slope of the graph when ? = 2

9) Find the intervals on t for which the curve is concave up/down

10) Find the tangent line to the curve at the point (3,6)

For problem 11-14 use the parametric equations ? = ? , ? = 2? − 1 to answer a-e.

11) Find ??/??

12) Find ?2?/??2

13) Find the slope of the graph when ? = ln5

14) Find the intervals on which the curve is concave up/down

15) Find the arc length along the curve defined by ? = 2? − 3 , ? = ?2, on the interval of t values [0,3]. Give an exact answer.

16) Find different polar coordinates for the point (7, 3?/2 ) that satisfy the conditions set forth in

a) ?<0

b) ? < 0

c) any set of coordinates not already used

17) Convert (3, 4?/3 ) from polar to rectangular coordinates.

18) Convert (−7,7√3) from rectangular to polar coordinates.

19) Convert ? = ?2 its corresponding polar equation.

20) Convert ? sin ? = 7 its corresponding rectangular equation.

21) Convert tan ? = 7 its corresponding rectangular equation.

22) Use calculus to find the area of contained by ? = 3 + 3 sin
?

23) Use calculus to find the area inside a single petal of ? = 5
cos 2?

24) Use calculus to find the area inside ?=6sin? outside
?=2+2sin?

25) Use calculus to find the area of the inner figure eight of ? = sin (?/2)

Answer #1

Graph the polar equations: r = 1 + cos θ and r = 1 + sin θ. Find
where they intersect (in polar or rectangular coordinates) and set
up the integral to find the area inside both curves?

Sketch the graph of the polar equation r = 3 + 2 sin
theta

Given parametric equations below, find d^2y/dx^2 and determine
the intervals on which the graph of the curve is concave up or
concave down.
(a) x = t^2 , y = t^3−3t
(b) x = cos(t), y = sin(2t)

6.) Let ~r(t) =< 3 cos t, -2 sin t > for 0 < t < pi.
a) Sketch the curve. Make sure to pay attention to the parameter
domain, and indicate the orientation of the curve on your graph. b)
Compute vector tangent to the curve for t = pi/4, and sketch this
vector on the graph.

1. Graph the curve given in parametric form by x = e t sin(t)
and y = e t cos(t) on the interval 0 ≤ t ≤ π2.
2. Find the length of the curve in the previous problem.
3. In the polar curve defined by r = 1 − sin(θ) find the points
where the tangent line is vertical.

a) Sketch the graph of r = 1 + sin2θ in polar coordinates with
proper explanation.
b) Find the area of the region that is inside of the cardioid r
= 2+2sinθ and outside of the circle r = 3. Also ﬁnd the area that
is outside of the cardioid and inside of the circle. Hence, ﬁnd the
total area enclosed by these two curves.

Graph the curve (? = −2 cos ? , ? = sin ? + sin 2?) . Then find
the point where the curve crosses itself and find the equation of
the two tangent lines at that point.

Eliminate the parameter to find a Cartesian equation of the
curve. Then sketch the curve and indicate with an arrow the
direction in which the curve is traced as the parameter
increases.
x = −3 + 2 cos(πt)
y = 1 + 2 sin(πt)
1 ≤ t ≤ 2

Sketch y = 2 sin(x) + x - 2
(b) This graph intersects the x-axis exactly once.
(i) Show that the x-intercept of the graph does not lie in the
interval (0, 0.7)
(ii) Find the the value of a, 0<a<1 for which the graph
will cross the x-axis in the interval (a, a + 0.1) and state this
interval
(iii) Given x = alpha is the root of the equations 2 Sin(x) + x
- 2 = 0, explain...

For the function ?(?) = ?^3 + 3x^2 + 1 determine
All intervals where the graph is concave up
All intervals where the graph is concave down
The coordinates of any points of inflection

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