Math Series 3
1. Determine the phase shift of y = 0.5cos(0.5x-π/3).
- π/3
- π/2
- 2π/5
- 2π/3
Answer: D
Solution:
Equate phase angle to zero then solve for x. $$y = 0.5 \cos \left( 0.5x-\frac{\pi}{3} \right)$$ $$\begin{align} 0.5x-\frac{\pi}{3} & = 0 \\ x & = \frac{2 \pi}{3} \\ \end{align}$$
Solution:
Equate phase angle to zero then solve for x. $$y = 0.5 \cos \left( 0.5x-\frac{\pi}{3} \right)$$ $$\begin{align} 0.5x-\frac{\pi}{3} & = 0 \\ x & = \frac{2 \pi}{3} \\ \end{align}$$
2. The function x3sinx is
- Odd
- Even
- Neither
- None
Answer: B
Solution:
Solution:
Even functions:
$$ f(x) = f(-x) $$ Odd functions: $$ -f(x) = f(-x) $$
Let x = 1.1 as test value.
$$\begin{align}
f(x) & = x^3 \sin(x) \\
\\
f(1.1) & = (1.1)^3 \sin(1.1)\\
& = 1.186 \\
\\
f(-1.1) & = (-1.1)^3 \sin(1.1)\\
& = 1.186 \\
\end{align}$$
Since $f(x) = f(-x)$, then the function is even.
$$ f(x) = f(-x) $$ Odd functions: $$ -f(x) = f(-x) $$
3. The function cos x + sin x + tan x + cot x + sec x + csc x is
- Both periodic and odd
- Both periodic and even
- Periodic but neither even nor odd
- Not periodic
Answer: C
Solution:
Let x = 1.1 as test value. $$\begin{align} f(x) & = \cos x + \sin x + \tan x + \cot x + \sec x + \csc x \\ f(1.1) & = 7.145 \\ f(-1.1) &= -1.829 \\ \end{align}$$ Since $f(x) \neq f(-x)$ and $-f(x) \neq f(-x)$ and all trigonometric functions are periodic, it is neither even nor odd.
Solution:
Let x = 1.1 as test value. $$\begin{align} f(x) & = \cos x + \sin x + \tan x + \cot x + \sec x + \csc x \\ f(1.1) & = 7.145 \\ f(-1.1) &= -1.829 \\ \end{align}$$ Since $f(x) \neq f(-x)$ and $-f(x) \neq f(-x)$ and all trigonometric functions are periodic, it is neither even nor odd.
4. The period of function sin 2x + cot 3x + sec 5x is
- π
- π/2
- 2π
- π/3
Answer: A
Solution:
$$ \sin 2x + \cot 3x + \sec 5x $$
$ \sin 2x$:
$$\begin{align}
T_1 & = \frac{2 \pi }{2}\\
& = \pi \\
\end{align}$$
$ \cot 3x$:
$$\begin{align}
T_2 & = \frac{3 \pi }{3}\\
& = \pi \\
\end{align}$$
$ \sec 5x$:
$$\begin{align}
T_3 & = \frac{2 \pi }{5}\\
\end{align}$$
The period of the function is the Least Common Multiple (LCM) of the periods of each term.
$$ LCM: T = \pi. $$
Hint: To get the LCM from the choices, divide the terms from the choices and the answer should be the least whole number.
$LCM: \frac{ \text{choices} }{ \text{given} } \equiv \text{whole number}$
Solution:
Function | Period |
---|---|
$y=A \sin (Bx+C)+K$ $y=A \sec (Bx+C)+K$ |
$T = \frac{2 \pi}{B}$ |
$y=A \cos (Bx+C)+K$ $y=A \csc (Bx+C)+K$ |
$T = \frac{2 \pi}{B}$ |
$y=A \tan (Bx+C)+K$ $y=A \cot (Bx+C)+K$ |
$T = \frac{\pi}{B}$ |
Hint: To get the LCM from the choices, divide the terms from the choices and the answer should be the least whole number.
$LCM: \frac{ \text{choices} }{ \text{given} } \equiv \text{whole number}$
5. Find the arc length of f(x) = x^2(3/2) + 2 on [1,4].
- 7.63
- 8.30
- 8.01
- 7.45
Answer: A
Solution:
Solution:
Arc Length - Rectangular
$$\begin{align}
\mathrm{S} & = \int_{x_1}^{x_2} \sqrt{ 1 + (dy/dx)^2 } dx \\
\mathrm{S} & = \int_{y_1}^{y_2} \sqrt{ 1 + (dx/dy)^2 } dy \\
\end{align}$$
Solving,
$$\begin{align}
y & = x^{3/2} + 2 \text{ } [1,4] \\
\frac{dy}{dx} & = \frac{3}{2} x^{1/2} \\
\\
\mathrm{S} & = \int_1^4 \sqrt{ 1 + \left( \frac{3}{2} x^{1/2} \right)^2 } dx\\
& = 7.63 \\
\end{align}$$
6. Compute the arc length of the curve by the parametric equations x = 1 + cos(t) and y = sin(t) where 0 ≤ t ≤ 2π.
- π
- 2π
- 2π/3
- π/2
Answer: B
Solution:
Solution:
Arc Length - Parametric
$$\begin{align}
\mathrm{S} & = \int_{t_1}^{t_2} \sqrt{ (dx/dt)^2 + (dy/dt)^2 } dt \\
\end{align}$$
Solving,
$$\begin{align}
x & = 1 + \cos(t) \\
dx/dt & = -\sin(t) \\
\\
y & = \sin(t)\\
dy/dt & = \cos(t)\\
\end{align}$$
$$\begin{align}
\mathrm{S} & = \int_0^{2 \pi} \sqrt{ (-\sin t)^2 + (\cos t)^2 } dt \\
& = 2 \pi. \\
\end{align}$$
7. Find the volume obtained if the region bounded by y=x^2 and y=2x is rotated about x-axis.
- 14.5
- 13.4
- 11.7
- 11.8
Answer: B
Solution:
$$\begin{align}
V & = 2 \pi \int_{y_1}^{y_2} rL dy \\
V & = 2 \pi \int_0^4 y \left( \sqrt{y} - \frac{y}{2} \right) dy \\
V & = 13.4. \\
\end{align}$$
Solution:
Using Shell Method
$$\begin{align}
V & = 2 \pi \int_{x_1}^{x_2} rL dx \\
V & = 2 \pi \int_{y_1}^{y_2} rL dy \\
\\
V & - \text{volume} \\
r & - \text{radius or the length from the axis of rotation to the strip}\\
L & - \text{length of the strip}\\
x_1, x_2 & - \text{points of intersection at x}\\
y_1, y_2 & - \text{points of intersection at y}\\
\end{align}$$
Graphing,
8. How far is the directrix of the parabola (x-4)^2 = -8(y-2) from the x-axis?
- x = 1
- x = -5
- x = 4
- x = 0
Answer: C
Solution:
As can be seen from the graph, distance = 4.
Solution:
Standard Equation of the Parabola
$$\begin{align}
(x-h)^2 & = \pm 4a(y-k) \\
(h,k) & - \text{vertex of the parabola} \\
a & - \text{focal length or the distance from} \\
& \text{ the vertex to the focus or to the directrix} \\
\end{align}$$
From the equation, $(x-4)^2 = -8(y-2)$, we can take that:
- The parabola opens downward since $(-)a$.
- The focal length is a = 2.
- The vertex is (4,2).
As can be seen from the graph, distance = 4.
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