How do you use the limit definition to find the slope of the tangent line to the graph #f(x)= 2x-x^2# at x=0?

Question

How do you use the limit definition to find the slope of the tangent line to the graph  #f(x)= 2x-x^2# at x=0?

Emma Aldridge · Accepted Answer

The slope of the tangent when #x=0# is #2# The slope of the tangent at any point is given by the derivative of the function at that point. The definition of the derivative of #y=f(x)# is # f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h # So with # f(x) = 2x-x^2 # and the value of #f'(0)# sought then; # f'(0)=lim_(h rarr 0) (f(h)-f(0))/h# # \ \ \ \ \ \ \ \ \=lim_(h rarr 0) ({2h-h^2}-{0-0})/h# # \ \ \ \ \ \ \ \ \=lim_(h rarr 0) (2h-h^2)/h# # \ \ \ \ \ \ \ \ \=lim_(h rarr 0) (2-h)# # \ \ \ \ \ \ \ \ \=2#

Isaiah Allen · Answer

undefined To find the slope of the tangent line to the graph of f(x) = 2x - x^2 at x = 0 using the limit definition, we can follow these steps:

1. Start with the equation of the function: f(x) = 2x - x^2.

2. Determine the equation of the tangent line. The equation of a line can be written as y = mx + b, where m represents the slope of the line. We need to find the slope (m) of the tangent line.

3. Use the limit definition of the derivative to find the slope. The derivative of a function f(x) at a specific point x = a can be defined as the limit of the difference quotient as h approaches 0. The difference quotient is given by [f(a + h) - f(a)] / h.

4. Substitute the values into the difference quotient. In this case, a = 0. So, we have [f(0 + h) - f(0)] / h.

5. Simplify the expression. Substitute the function f(x) = 2x - x^2 into the difference quotient and simplify the numerator.

6. Take the limit as h approaches 0. Evaluate the expression as h approaches 0 to find the slope of the tangent line.

7. The resulting value will be the slope of the tangent line to the graph of f(x) = 2x - x^2 at x = 0.