Question

Let f(x)=1/4+3 x and point P=(1, 1/7)

a. Use the following definition of the slope of the tangent line at x=a to find the slope of the line tangent to the to the graph off at P
mₜₐₙ=limₕ→ 0 f(a+h)-f(a)/h

Answer 1

The slope of the tangent line to the graph of f(x) at point P(1, 1/7) is calculated using the limit definition of the derivative, resulting in a slope of 3.

Step-by-step explanation:

To find the slope of the tangent line to the graph of the function f(x) at the point P(1, 1/7), we will apply the definition of the slope of the tangent line at x=a, which is mt = limh→0 (f(a+h)-f(a))/h. Since the given function is f(x) = 1/4 + 3x, we calculate the slope at a = 1 as follows:

1. First, evaluate f(a) = f(1) = 1/4 + 3(1) = 1/4 + 3 = 31⁄4.
2. Then evaluate f(1+h) which becomes 1/4 + 3(1+h).
3. Using the definition, mt = limh→0 [(1/4 + 3(1+h)) - (31⁄4)]/h.
4. Simplify this expression: mt = limh→0 (3h)/h.
5. As h approaches 0, the h terms cancel resulting in the slope of the tangent line being 3.

Thus, the slope at P is 3, which is also the derivative of f(x) at x = 1.