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{:[f^(')(x)=9e^(x)" and "],[f(8)=-8+9e^(8).],[f(0)=◻]:}

f′(x)=9ex and f(8)=−8+9e8.f(0)=□ \begin{array}{l}f^{\prime}(x)=9 e^{x} \text { and } f(8)=-8+9 e^{8} . \\ f(0)=\square\end{array}

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Full solution

Q. f′(x)=9ex and f(8)=−8+9e8.f(0)=□ \begin{array}{l}f^{\prime}(x)=9 e^{x} \text { and } f(8)=-8+9 e^{8} . \\ f(0)=\square\end{array}
  1. Integrate derivative to find function: We are given the derivative of the function f(x)f(x) as f′(x)=9exf'(x) = 9e^x. We need to find the original function f(x)f(x). To do this, we integrate the derivative f′(x)f'(x).\newlineIntegration of f′(x)=9exf'(x) = 9e^x gives us f(x)=9ex+Cf(x) = 9e^x + C, where CC is the constant of integration.
  2. Find constant of integration: We are given the value of the function at x=8x = 8, which is f(8)=−8+9e8f(8) = -8 + 9e^8. We can use this information to find the constant of integration CC. Substitute x=8x = 8 into f(x)f(x) to get f(8)=9e8+Cf(8) = 9e^8 + C. −8+9e8=9e8+C-8 + 9e^8 = 9e^8 + C Solve for CC: C=−8+9e8−9e8C = -8 + 9e^8 - 9e^8 C=−8C = -8
  3. Complete function: Now that we have the constant of integration, we can write the complete function f(x)f(x) as f(x)=9ex−8f(x) = 9e^x - 8.
  4. Find f(0)f(0): To find f(0)f(0), we substitute x=0x = 0 into the function f(x)f(x).
    f(0)=9e0−8f(0) = 9e^0 - 8
    Since e0=1e^0 = 1, we have f(0)=9(1)−8f(0) = 9(1) - 8
    f(0)=9−8f(0) = 9 - 8
    f(0)=1f(0) = 1

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