There are two neighboring points P and Q on the curve f(x)=x^2
(a) Determine m(h), the slope of the straight line PQ
(b) The slope, S, of the tangent at P is defined as S=lim h->0 m(h). Find the value of S.
(c) Use this value of S to show that the equation of the tangent at P is y=4x-4.
To solve these questions we need to use the concept of the derivative. The derivative of a function measures the rate at which the function is changing at a particular point. In this case, we want to find the slope of the tangent line at point P.
(a) To determine the slope of the straight line PQ, we need to find the coordinates of points P and Q. Let's assume that P is located at (a, f(a)), and Q is located at (a+h, f(a+h)). Substituting these points into the equation f(x)=x^2, we get:
P: (a, f(a)) = (a, a^2)
Q: (a+h, f(a+h)) = (a+h, (a+h)^2)
The slope, m(h), is defined as the change in y divided by the change in x:
m(h) = (f(a+h) - f(a)) / (a+h - a)
= ((a+h)^2 - a^2) / h
= (a^2 + 2ah + h^2 - a^2) / h
= (2ah + h^2) / h
= 2a + h
Therefore, the slope of the straight line PQ, m(h), is 2a + h.
(b) Now, let's find the slope, S, of the tangent at P by taking the limit of m(h) as h approaches 0:
S = lim(h->0) m(h)
= lim(h->0) (2a + h)
= 2a + 0
= 2a
Hence, the slope of the tangent at P is 2a.
(c) To show that the equation of the tangent at P is y = 4x - 4, we need to find the point of tangency and the slope of the tangent. We already know that the slope of the tangent at P is 2a.
Substitute the coordinates of P, which is (a, f(a)), into the equation of a line in point-slope form, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
Plugging in the values, we get:
y - f(a) = 2a(x - a)
y - a^2 = 2ax - 2a^2
y = 2ax - 2a^2 + a^2
y = 2ax - a^2
Comparing this equation with y = 4x - 4, we can see that 2a = 4 and -a^2 = -4.
From the equation 2a = 4, we can solve for a:
2a = 4
a = 2
Substituting this value of a into the equation for y, we get:
y = 2(2)x - (2)^2
y = 4x - 4
Hence, the equation of the tangent at P is y = 4x - 4.