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  1. Asked: March 28, 2021In:

    Slope of a Function

    acedstud
    acedstud

    Any slope of a function is given by the rise divided by the run. Any function where the rise is equal to the run will have a slope equal to 1. The equation of a straight line is of the form $y=mx+c$, where $m$ is the slope and $c$ is the $x$-intercept. So, any function of the form $y=x+c$, where $c$Read more

    Any slope of a function is given by the rise divided by the run. Any function where the rise is equal to the run will have a slope equal to 1.

    The equation of a straight line is of the form y=mx+c, where m is the slope and c is the x-intercept.

    So, any function of the form y=x+c, where c is any real number will have a slope of 1.

    Examples: y=x, y=x+4, y=x-3, etc.

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  2. Asked: March 28, 2021In:

    What is Zero Factorial equal to

    acedstud
    acedstud

    Zero factorial (0!) is equal to 1 by the empty-product convention. The empty-product convention is the result of multiplying no factors and is equal to the multiplicative identity which is 1.

    Zero factorial (0!) is equal to 1 by the empty-product convention.

    The empty-product convention is the result of multiplying no factors and is equal to the multiplicative identity which is 1.

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  3. Asked: March 28, 2021In: ,

    What is integral function

    acedstud
    acedstud

    An integral function is a function defined by integration. The integral of a function $y=f(x)$ is the area under the curve of $f(x)$. That is the area of the portion bounded by the curve, $f(x)$, and the $x$-axis.

    An integral function is a function defined by integration.

    The integral of a function y=f(x) is the area under the curve of f(x).

    That is the area of the portion bounded by the curve, f(x), and the x-axis.

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  4. Asked: March 28, 2021In: ,

    what is crammer rule?

    acedstud
    Best Answer
    acedstud

    Cramer's rule is a method of solving a system of linear equations using determinants of matrices.

    Cramer’s rule is a method of solving a system of linear equations using determinants of matrices.

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  5. Asked: March 27, 2021In: ,

    Fourier Analysis

    acedstud
    acedstud
    This answer was edited.

    The given periodic function can be rewritten as $f(t)=\begin{cases} 1 & \text{ if } 0<t<0.2 \\ 3 & \text{ if } 0.2<t<0.4 \end{cases}$ $f(t+0.4)=f(t)$ The Fourier series expansion of a function $f(t)$ with period, $T$, is given by $f(t)=\dfrac{a_0}{2}+\sum_{n=1}^{\infty}{\left\{a_Read more

    The given periodic function can be rewritten as

    f(t)=\begin{cases} 1 & \text{ if } 0<t<0.2 \\ 3 & \text{ if } 0.2<t<0.4 \end{cases}
    f(t+0.4)=f(t)

    The Fourier series expansion of a function f(t) with period, T, is given by f(t)=\dfrac{a_0}{2}+\sum_{n=1}^{\infty}{\left\{a_n\cos{\dfrac{2n\pi t}{T}}+b_n\sin{\dfrac{2n\pi t}{T}}\right\}}, where a_0=\dfrac{2}{T}\int_{0}^{T}f(t)dt, a_n=\dfrac{2}{T}\int_{0}^{T}f(t)\cos{\dfrac{2n\pi t}{T}}dt, and b_n=\dfrac{2}{T}\int_{0}^{T}f(t)\sin{\dfrac{2n\pi t}{T}}dt

    Here, T=0.4, so we have: f(t)=\dfrac{a_0}{2}+\sum_{n=1}^{\infty}{\{a_n\cos{5n\pi t}+b_n\sin{5n\pi t}\}}.

    Where a_0=\dfrac{2}{T}\int_{0}^{T}f(t)dt=5\int_{0}^{0.4}f(t)dt

    =5\left(\int_{0}^{0.2}dt+\int_{0.2}^{0.4}3dt\right)=5\left([t]_0^{0.2}+[3t]_{0.2}^{0.4}\right)

    =5(0.2+0.6)=4.

    a_n=\dfrac{2}{T}\int_{0}^{T}f(t)\cos{\dfrac{2n\pi t}{T}}dt=5\int_{0}^{0.4}f(t)\cos{5n\pi t}dt

    =5\left(\int_{0}^{0.2}\cos{5n\pi t}dt+\int_{0.2}^{0.4}3\cos{5n\pi t}dt\right)

    =5\left(\left[\dfrac{\sin{5n\pi t}}{5n\pi}\right]_0^{0.2}+3\left[\dfrac{\sin{5n\pi t}}{5n\pi}\right]_{0.2}^{0.4}\right)

    =0        [\because \sin{n\pi}=0].

    b_n=\dfrac{2}{T}\int_{0}^{T}f(t)\sin{\dfrac{2n\pi t}{T}}dt=5\int_{0}^{0.4}f(t)\sin{5n\pi t}dt

    =5\left(\int_{0}^{0.2}\sin{5n\pi t}dt+\int_{0.2}^{0.4}3\sin{5n\pi t}dt\right)

    =5\left(-\left[\dfrac{\cos{5n\pi t}}{5n\pi}\right]_0^{0.2}-3\left[\dfrac{\cos{5n\pi t}}{5n\pi}\right]_{0.2}^{0.4}\right)

    =-\dfrac{\cos n\pi}{n\pi}+\dfrac{1}{n\pi}-\dfrac{3\cos2n\pi}{n\pi}+\dfrac{3\cos n\pi}{n\pi}

    =\dfrac{2\cos n\pi}{n\pi}-\dfrac{2}{n\pi}

    =0 (when n is even), and \dfrac{-4}{n\pi} (when n is odd).

    Therefore, the Fourier series is given by f(t)=2-\dfrac{4}{\pi}\left\{\sin5\pi t+\dfrac{1}{3}\sin15\pi t+\dfrac{1}{5}\sin25\pi t+... \right \}.

     

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  6. Asked: March 27, 2021In:

    Question of the Day (27/03/2021)

    acedstud
    acedstud

    a.) The mean/average of a set of data is given by the total score divided by the count of the data. Thus, $\text{Mean}=\dfrac{22+22+19+31+36+50+38+25+30}{9}=\dfrac{273}{9}=30.33$.   b.) The median is the score at the center when the set of data is arranged in order. Thus, arranging the set of dRead more

    a.) The mean/average of a set of data is given by the total score divided by the count of the data.

    Thus, \text{Mean}=\dfrac{22+22+19+31+36+50+38+25+30}{9}=\dfrac{273}{9}=30.33.

     

    b.) The median is the score at the center when the set of data is arranged in order.

    Thus, arranging the set of data, we have: 19, 22, 22, 25, 30, 31, 36, 38, 50.

    Therefore, the median is 30.

     

    c.) The mode is the most occurring score. 22 occurred twice whereas other numbers occurred once.

    Therefore, the mode is 22.

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  7. Asked: March 26, 2021In: ,

    Question of the Day (26/03/2021)

    acedstud
    acedstud
    This answer was edited.

    A zero of a function is a value of the function when the function equals zero. For example, given the function $f(x)$, the zeros of $f(x)$ are the values of $x$ when $f(x)=0$. 2. (a) For the function $y=3x^2+2x-5$, the zeros of the function are the value of $x$, when $3x^2+2x-5=0$. Solving $3x^2+2x-Read more

    1. A zero of a function is a value of the function when the function equals zero. For example, given the function f(x), the zeros of f(x) are the values of x when f(x)=0.

    2.

    (a) For the function y=3x^2+2x-5, the zeros of the function are the value of x, when 3x^2+2x-5=0.

    Solving 3x^2+2x-5=0 by grouping and factorizing gives 3x^2-3x+5x-5=0\Rightarrow(3x^2-3x)+(5x-5)=0

    \Rightarrow3x(x-1)+5(x-1)=0\Rightarrow(3x+5)(x-1)=0

    \Rightarrow3x+5=0 or x-1=0

    \Rightarrow x=-\dfrac{5}{3} or x=1.

    Therefore, the zeros of y=3x^2+2x-5 are x=-\dfrac{5}{3} and x=1

     

    (b) For the function y=4^{3x-9}-1, the zeros of the function are the value of x, when 4^{3x-9}-1=0 or 4^{3x-9}=1.

    Recall that 4^0=1.

    So, we have 4^{3x-9}=4^0

    Equating the exponents, we have 3x-9=0\Rightarrow3x=9\Rightarrow x=\dfrac{9}{3}=3.

    Therefore, the zero of y=4^{3x-9}-1 is at x=3.

     

    (c) For the function y=\sin(2x)+1; 0\le x\le2\pi, the zeros of the function are the value of x between 0 and 2\pi, where \sin(2x)+1=0 or \sin(2x)=-1.

    \sin(2x)=-1\Rightarrow2x=\sin^{-1}(-1)=\dfrac{3\pi}{2}+2n\pi, n=0, 1, 2, …

    \Rightarrow x=\dfrac{1}{2}\left(\dfrac{3\pi}{2}+2n\pi\right)=\dfrac{3\pi}{4}+n\pi

    For n = 0, x=\dfrac{3\pi}{4}

    For n =1, x=\dfrac{3\pi}{4}+\pi=\dfrac{7\pi}{4}

    For n =2, x=\dfrac{3\pi}{4}+2\pi=\dfrac{11\pi}{4} which is greater than 2\pi.

    Therefore, the zeros of y=\sin(2x)+1; 0\le x\le2\pi are \dfrac{3\pi}{4} and \dfrac{7\pi}{4}.

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  8. Asked: March 24, 2021In: ,

    Methods of Factorization

    acedstud
    acedstud

    There are many methods of factorisation. The three commonest types are: 1) Factorisation by factoring out the GCD. Example: Factor $3x^2+6x$. Here, the GCD (the greatest factor both terms have in common) is $3x$. So, factoring out $3x$ gives $3x(x+2)$. Therefore, $3x^2+6x=3x(x+2)$.   2) FactoriRead more

    There are many methods of factorisation.

    The three commonest types are:

    1) Factorisation by factoring out the GCD.

    Example: Factor 3x^2+6x.

    Here, the GCD (the greatest factor both terms have in common) is 3x. So, factoring out 3x gives 3x(x+2).

    Therefore, 3x^2+6x=3x(x+2).

     

    2) Factorisation by grouping.

    Example: to factor 3x^3+2x^2+9x+6, we group it as follows: x^2(3x+2)+3(3x+2)=(x^2+3)(3x+2).

    Therefore, 3x^3+2x^2+9x+6=(x^2+3)(3x+2).

     

    3.) Factoring quadratic trinomials by trial and error.

    Example: To factor x^2+5x+6, we use the formula x^2+(a+b)x+ab=(x+a)(x+b). So, here, we have x^2+5x+6=x^2+(2+3)x+2\times3=(x+2)(x+3).

    Therefore, x^2+5x+6=(x+2)(x+3).

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  9. Asked: March 23, 2021In: ,

    Question of the Day (23/03/2021)

    acedstud
    acedstud
    This answer was edited.

    The future value (FV) of an investment (P) at an annual compound interest rate of (r) compounded $t$ times a year for $n$ years is given by $FV=P\left(1+\dfrac{r}{t}\right)^{nt}$ Here: $P=$\$$8,000, r=8\%=0.08, n=9\text{ months}=\dfrac{9}{12}\text{ years}$. Since the CD is compounded monthly, that mRead more

    The future value (FV) of an investment (P) at an annual compound interest rate of (r) compounded t times a year for n years is given by

    FV=P\left(1+\dfrac{r}{t}\right)^{nt}

    Here: P=$8,000, r=8\%=0.08, n=9\text{ months}=\dfrac{9}{12}\text{ years}.

    Since the CD is compounded monthly, that means that in a year it will be compounded 12 times, so t=12.

    Substituting all values into the formula, we have:

    FV=8000\left(1+\dfrac{0.08}{12}\right)^{\frac{9}{12}\times12}

    =8000(1+0.006667)^9=8000(1.006667)^9

    =8000\times1.061625

    =8,493.00

    Therefore, the value of her investment at the maturity of the CD is $8,493.00.

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  10. Asked: March 22, 2021In:

    Equivalent Fractions

    acedstud
    acedstud
    This answer was edited.

    Equivalent fractions are fractions which when simplified to the lowest term give the same fraction. Equivalent fractions are obtained by multiplying/dividing the numerator and the denominator of a fraction by a common factor. Given the fraction $\dfrac{4}{5}$, multiplying the numerator and the denomRead more

    Equivalent fractions are fractions which when simplified to the lowest term give the same fraction. Equivalent fractions are obtained by multiplying/dividing the numerator and the denominator of a fraction by a common factor.

    Given the fraction \dfrac{4}{5}, multiplying the numerator and the denominator by 2 gives \dfrac{4\times2}{5\times2}=\dfrac{8}{10}.

    Multiplying the numerator and the denominator by 3 gives \dfrac{4\times3}{5\times3}=\dfrac{12}{15}.

    Multiplying the numerator and the denominator by 4 gives \dfrac{4\times4}{5\times4}=\dfrac{16}{20}.

    Thus, \dfrac{8}{10}, \dfrac{12}{15}, \dfrac{16}{20} are equivalent fractions to \dfrac{4}{5}.

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