4.3: Chapter 4 Review Exercises

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Apr 1, 2025
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- 4.2E: Exercises for Section 4.2
- 5: Introduction to Differential Equations

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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

$y=1−x^3$

$Graph of a curve starting at (1, 0) and decreasing into the fourth quadrant.$

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

Answer

$\dfrac{x^2}{16}+(y−1)^2=1$

$Graph of an ellipse with center (0, 1), major axis horizontal and of length 8, and minor axis of length 2.$

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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Exercises 1 - 4

Exercises 5 -8

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