4.3: Chapter 4 Review Exercises

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Page ID: 176343

Doli Bambhania, Neelam R. Shukla, and Danny Tran
De Anza College

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Exercises 1 - 4

True or False? Justify your answer with a proof or a counterexample.

The rectangular coordinates of the point $\left(4,\frac{5π}{6}\right)$ are $\left(2\sqrt{3},−2\right).$
The equations $x=\cosh(3t), \; y=2\sinh(3t)$ represent a hyperbola.

Answer: True

The arc length of the spiral given by $r=\dfrac{θ}{2}$ for $0≤θ≤3π$ is $\frac{9}{4}π^3$ units.
Given $x=f(t)$ and $y=g(t)$, if $\dfrac{dx}{dy}=\dfrac{dy}{dx}$, then $f(t)=g(t)+C,$ where $C$ is a constant.

Answer: False. Imagine $y=t+1, \; x=−t+1.$

Exercises 5 -8

Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.

$x=1+t, \; y=t^2−1, \quad −1≤t≤1$
$x=e^t, \; y=1−e^{3t}, \quad 0≤t≤1$

Answer: $y=1−x^3$

$Graph of a curvet. Details in the caption.$
Figure $\PageIndex{1}$: Graph of a curve starting at (1, 0) and decreasing into the fourth quadrant.

$x=\sin θ, \; y=1−\csc θ, \quad 0≤θ≤2π$
$x=4\cos ϕ, \; y=1−\sin ϕ, \quad 0≤ϕ≤2π$

Answer: $\dfrac{x^2}{16}+(y−1)^2=1$

$Graph of an ellipse with center (0, 1), details in the caption.$
Figure $\PageIndex{2}$: Graph of an ellipse with center (0, 1), major axis horizontal and of length 8, and minor axis of length 2.

Find the equation of the tangent line to the given curve. Graph both the function and its tangent line.

$x=\ln(t),\; y=t^2−1, \; t=1$

Find $\dfrac{dy}{dx}, \; \dfrac{dx}{dy},$ and $\dfrac{d^2x}{dy^2}$ of $y=(2+e^{−t}), \; x=1−\sin t$

Find the area of the region.

$x=t^2, \; y=\ln(t), \quad 0≤t≤e$

Answer: $\dfrac{e^2}{2}\text{ units}^2$

Find the arc length of the curve over the given interval.

$x=3t+4, \; y=9t−2, \quad 0≤t≤3$

Answer: $9\sqrt{10}$ units

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