the derivative of a function f is given by f(x)=0.1x + e^0.25x. at what value of x for x > 0 does the line…

the derivative of a function f is given by f(x)=0.1x + e^0.25x. at what value of x for x > 0 does the line tangent to the graph of f at x have slope 2? a 0.512 b 1.849 c 2.287 d 8.113

the derivative of a function f is given by f(x)=0.1x + e^0.25x. at what value of x for x > 0 does the line tangent to the graph of f at x have slope 2? a 0.512 b 1.849 c 2.287 d 8.113

Answer

Explanation:

Step1: Set the derivative equal to slope

We know that the slope of the tangent line is given by the derivative. So we set $f'(x)=2$, which gives the equation $0.1x + e^{0.25x}=2$.

Step2: Use numerical methods

This equation $0.1x + e^{0.25x}-2 = 0$ cannot be solved algebraically. We can use a graphing - calculator or a numerical method like Newton - Raphson method. Let $g(x)=0.1x + e^{0.25x}-2$, then $g'(x)=0.1 + 0.25e^{0.25x}$. Starting with an initial guess (for example, $x = 1$), and using the Newton - Raphson formula $x_{n + 1}=x_{n}-\frac{g(x_{n})}{g'(x_{n})}$. Or using a graphing calculator's solve function, we find the root of the equation $0.1x + e^{0.25x}-2 = 0$ for $x>0$.

Answer:

B. 1.849