write the equation of the tangent line to the graph of the function at the given point. 7.) f(x)=x + e^x…

write the equation of the tangent line to the graph of the function at the given point. 7.) f(x)=x + e^x, (0,1) 8.) g(t)=sin t+\\frac{1}{2 - e^t}, (\\frac{\\pi}{2},\\frac{1}{2 - e^{\\pi}})

write the equation of the tangent line to the graph of the function at the given point. 7.) f(x)=x + e^x, (0,1) 8.) g(t)=sin t+\\frac{1}{2 - e^t}, (\\frac{\\pi}{2},\\frac{1}{2 - e^{\\pi}})

Answer

Explanation:

Step1: Find the derivative of the function

For (y = f(x)=x + e^{x}), using the sum - rule and the derivative formulas (\frac{d}{dx}(x)=1) and (\frac{d}{dx}(e^{x})=e^{x}), we have (y'=f'(x)=1 + e^{x}).

Step2: Evaluate the derivative at the given (x) - value

The point is ((0,1)), when (x = 0), (f'(0)=1+e^{0}=1 + 1=2). The slope (m) of the tangent line is (2).

Step3: Use the point - slope form of a line

The point - slope form is (y - y_{1}=m(x - x_{1})), where ((x_{1},y_{1})=(0,1)) and (m = 2). Substituting these values, we get (y-1=2(x - 0)).

Step4: Simplify the equation

(y-1=2x), so (y = 2x+1).

Answer:

(y = 2x + 1)