---### Problem 1:**Objective:**Find a Taylor series for \( f(x) = e^{5x} \) centered at \( c = -3 \). Show your work neatly and detailed in the style of the lecture notes for full credit.**Given Function:**\[ f(x) = e^{5x} \]**Taylor Series Formula:**The Taylor series of the function \( f(x) \) centered at \( c \) is given by:\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(c)}{k!} (x - c)^k \]1. **Compute \( f(-3) \):** \[ f(-3) = e^{5(-3)} = e^{-15} \]2. **Find the first derivative of \( f(x) \) and its value at \( c = -3 \):** \[ f'(x) = 5e^{5x} \] \[ f'(-3) = 5e^{5(-3)} = 5e^{-15} \](Note: The content should include additional higher-order derivatives and their values properly computed, extending this methodology.)---The above content explains the initial steps to determine the Taylor series of the function e^5x centered at -3. Following the outlined steps, one would need to compute higher derivatives and their values at -3, then substitute these into the Taylor series formula.

Name: ---### Problem 1:**Objective:**Find a Taylor series for \( f(x) = e^{
Uploaded: 2024-03-20T07:07:26.000Z
Channel: Bartleby
Description: ---### Problem 1:**Objective:**Find a Taylor series for \( f(x) = e^{5x} \) centered at \( c = -3 \). Show your work neatly and detailed in the style of the lecture notes for full credit.**Given Function:**\[ f(x) = e^{5x} \]**Taylor Series Formula: