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Math Help - calculus with exponentials

  1. #1
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    calculus with exponentials

    simplify: exp(a)^-1 * exp (b)^2/exp(5c)

    calculate: d/dx exp(xsin(x))

    calculate: (integral)xexp(x^2-3)dx

    find the exponential function f(x)=a^x whose graph goes through point 5, 1/32. a=???

    find the exponential funtion f(x)=Ca^x whose graph goes through points (0,2) and (4,32). a=??? C=???
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    Quote Originally Posted by keemariee View Post
    simplify: exp(a)^-1 * exp (b)^2/exp(5c)

    Iso: You just need to know basic laws of exponentiation. Like e^x e^y = e^{x+y} and e^{-x} = \frac1{e^x}

    So your question reads
    (e^{a})^{-1} {(e^b)}^2/e^{5c} = e^{-a + 2b -5c}

    calculate: d/dx exp(xsin(x))

    Iso: Use chain rule:
    \frac{d(e^{x \sin x})}{dx} = e^{x\sin x} \frac{d(x\sin x)}{dx}=....

    So can you continue now?


    calculate: (integral)xexp(x^2-3)dx

    Iso: Substitute u = x^2 - 3, so that du = 2x \, dx

    \boxed{\int x e^{x^2-3}\,dx = \frac12 \int e^{u}\,du = \frac{e^{x^2 - 3}}{2}}

    find the exponential function f(x)=a^x whose graph goes through point 5, 1/32. a=???

    find the exponential funtion f(x)=Ca^x whose graph goes through points (0,2) and (4,32). a=??? C=???
    ..
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    Hi, keemariee! I've rewritten the expressions in your problems, so let me know if I misinterpreted anything.

    Quote Originally Posted by keemariee View Post
    simplify: \frac{\text{e}^{-a}\,\text{e}^{2b}}{\text{e}^{5c}}
    Use these properties:

    a^ma^n = a^{m+n}

    \frac{a^m}{a^n} = a^{m-n}

    Quote Originally Posted by keemariee View Post
    calculate: \frac{d}{dx} \left[\text{e}^{x\sin x}\right]
    Use the chain rule:

    \frac{d}{dx} \left[\text{e}^{x\sin x}\right]

    =\text{e}^{x\sin x}\frac{d}{dx}\left[x\sin x\right]

    And now apply the product rule.

    Quote Originally Posted by keemariee View Post
    calculate: \int x\,\text{e}^{x^2-3}\,dx
    Use substitution if you have to (letting u = x^2 - 3), or just bring in a factor of 2 and recognize that

    2x\,\text{e}^{x^2 - 3} = \text{e}^{x^2 - 3}\frac{d}{dx}\left[\text{e}^{x^2 - 3}\right]

    You can apply the chain rule in reverse.

    Quote Originally Posted by keemariee View Post
    find the exponential function f(x)=a^x whose graph goes through point \left(5,\ \frac1{32}\right). a=\text{???}
    Substitute the values:

    f(5) = \frac1{32}\Rightarrow a^5 = \frac1{32}

    Now solve for a:

    a = \sqrt[5]{\frac1{32}} = \frac1{\sqrt[5]{32}}

    What value can you raise to the 5th power to get 32?

    Quote Originally Posted by keemariee View Post
    find the exponential funtion f(x)=Ca^x whose graph goes through points (0,\ 2) and (4,\ 32). a=\text{???},\ C=\text{???}
    Do the same thing as the last problem. The only difference is that you will have two equations with two unknowns, so you will need to solve a system.
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  4. #4
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    Hello,

    Quote Originally Posted by keemariee View Post
    find the exponential function f(x)=a^x whose graph goes through point 5, 1/32. a=???
    We know it goes through (5,1/32)

    --> f(5)=a^{5}=\frac{1}{32}

    But 32=2^5 \implies a=\frac 12

    find the exponential funtion f(x)=Ca^x whose graph goes through points (0,2) and (4,32). a=??? C=???
    Again :
    2=f(0)=Ca^0 \implies \boxed{C=2}

    f(4)=32 \implies 32=2 \cdot a^4

    a^4=16

    But 16=2^4 \implies a=2
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    Quote Originally Posted by keemariee View Post
    find the exponential function f(x)=a^x whose graph goes through point 5, 1/32. a=???
    \frac1{32} = f(5) = a^5 \Rightarrow  a^5 = \frac1{32} = \left(\frac12 \right)^5 \Rightarrow a = \frac12


    Quote Originally Posted by keemariee View Post
    find the exponential funtion f(x)=Ca^x whose graph goes through points (0,2) and (4,32). a=??? C=???
    2 = f(0) = Ca^0 = C

    32 = f(4) = Ca^4 \Rightarrow 2a^4 = 32 \Rightarrow a^4 = 16 \Rightarrow a = \pm2

    So actually there are two graphs...
    f(x) = 2(-2)^x and f(x) = 2^{x+1}
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