let f be the function given by f(x)=-2e^x^3. F0r what value of x is the slope of the line tangent to the graph of f at (x,f(x)) equal to -6
let f(u)=e^u
and u= x^3
f'(u)=e^u du/dx
but du/dx= 3x^2
f'(x)=e^(x^3) * 3x^2
so, now you just want to find where
-2*3x^2 e^x^3 = -6
x^2 e^x^3 = 1
I think a graphical solution will be easiest.
To find the value of x where the slope of the tangent line to the graph of f(x) is equal to -6, we need to find the derivative of the function f(x) and then solve for x.
Step 1: Find the derivative of f(x)
The derivative of -2e^x^3 can be found using the chain rule. The chain rule states that if you have a function of the form f(g(x)), then its derivative is f'(g(x)) * g'(x).
In this case, the function f(x) can be written as f(x) = -2e^(x^3). So, let's apply the chain rule:
f'(x) = (-2) * (e^(x^3))' [Applying the chain rule]
= (-2) * (3x^2) * (e^(x^3)) [Differentiating e^(x^3) using the chain rule]
= -6x^2 * e^(x^3)
Step 2: Set the derivative equal to -6 and solve for x
We know that the slope of the tangent line is equal to the derivative, so we need to find the value of x where the derivative is -6. Let's set up the equation:
-6x^2 * e^(x^3) = -6
Now, we can divide both sides of the equation by -6 to simplify it:
x^2 * e^(x^3) = 1
Step 3: Solve for x
Since there is no easy algebraic way to solve this equation directly, we can use numerical methods or approximation techniques to find the solution. One approach is to use a graphing calculator or software to plot the function y = x^2 * e^(x^3) - 1 and locate its x-intercept (where the y-value is zero). Alternatively, you can use iterative methods such as the Newton-Raphson method or the bisection method to estimate the solution.
By solving the equation x^2 * e^(x^3) = 1, you can find the value of x where the slope of the tangent line to the graph of f(x) is equal to -6.