Suppose the function
a) Use the definition of the derivative to show that f ' (-2) = -1.
b) Write an equation for the line tangent to the graph of f at x = -2.
To find the derivative and equation of the tangent line, we need to know the function f(x). Without that information, it is not possible to provide a specific answer. However, I can explain the general steps involved in finding the derivative and equation of the tangent line for any given function.
a) To use the definition of the derivative, we need to know the function f(x). The derivative of a function f'(x) represents the rate of change of the function at each point. In this case, we are asked to find f'(-2), which means we want to find the instantaneous rate of change (slope) of the graph of f(x) at x = -2.
1. Begin by determining the function f(x), as it is essential in finding the derivative.
2. Apply the definition of the derivative, which states that f'(x) = lim(h â 0) [(f(x+h) - f(x)) / h].
3. Plug in the value x = -2 into the derivative formula and evaluate the limit. Simplify the expression to solve for f'(-2). If f'(-2) evaluates to -1, it means the slope of the tangent line at x = -2 is -1.
b) After finding the derivative f'(-2), we can write the equation of the tangent line using the point-slope form. The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
1. Use the given point (x = -2, y = f(-2)), which lies on the graph of f, to find the y-coordinate for the tangent line. Replace x with -2 in the function f(x) to compute the value of f(-2).
2. With the slope f'(-2) from part a), substitute the value of m in the point-slope form equation.
3. Replace x with (x - x1) and y with (y - y1), where x1 and y1 are -2 and f(-2), respectively.
4. Simplify and rearrange the equation to get the final equation of the tangent line.
Remember, the specific steps and calculations will depend on the given function f(x).