Final answer:
The slope of the tangent line to the function f(x) = 5 - 5x^2 at P(0, 5) is determined by finding the derivative, which in this case is f'(x) = -10x. Substituting x = 0 gives a slope of 0. Therefore, the equation of the tangent line at P is y = 5.
Step-by-step explanation:
To find the slope of the tangent line to the graph of a function at a given point, we utilize the definition of the derivative as the limit of the average rate of change as the interval approaches zero.
For the function f(x) = 5 - 5x^2 at the point P(0, 5), we first need to determine the derivative of the function:
f'(x) = lim(h→0) [f(a+h) - f(a)] / h
Now applying this to our function:
f'(x) = -10x
To find the slope at P(0, 5), we substitute x = 0 into the derivative:
f'(0) = -10(0) = 0
Thus, the slope of the curve at P is 0, meaning the tangent line is horizontal.
To determine the equation of the tangent line at P, we use the point-slope form, with the point (0, 5) and slope 0:
y - y1 = m(x - x1)
y - 5 = 0(x - 0)
y = 5
The equation of the tangent line at P is y=5.
Complete Question:
a. Use the definition mtan= limh→0 f(a+h)−f(a)/ h to find the slope of the line tangent to the graph of f at P.
b. Determine an equation of the tangent line at P.
f(x)=5−5x^2; P(0,5)