Let (a, b) be any point on the graph of Prove that the area of the

triangle formed by the tangent through (a, b) and the coordinate axes is 2.

This is the solution from the solution manual
The coordinates of the point are (a,1/a). The slope
of the tangent is -1/a^2. The equation of the tangent
is y-1/a=-1/a^2(x-a) or y=(-1/a^2)x + 2/a. The
intercepts are (0,2/a) and (-2a,0). The tangent line
and the axes form a right triangle with legs of length 2/a and 2a. The area of the triangle is (1/2)(2/a)(2a)=2.

I understand everything about this except I don't know how the slope of the tangent is -1/a^2 was calculated. Also, I'm confused about the area of the triangle equation. Shouldn't it be Area=(height x base)/2 so ((2/a)(2a))/2 which is different from (1/2)(2/a)(2a)=2.

if f(x) = 1/x, then since

1/x = x^-1, f' = (-1)x^-2 = -1/x^2

at x=1, that is -1/a^2

huh? ((2/a)(2a))/2 = (1/2)(2/a)(2a)
dividing by 2 is the same as multiplying by 1/2, right?

6. The following are the x-intercepts, EXCEPT

a. -5 b. c. d. 2

7. Which of the following is the y-intercept?

a. -120 b. -60 c. 60 d. 120

To understand how the slope of the tangent (-1/a^2) was calculated, let's start by finding the derivative of the function that represents the curve at the point (a, b).

Let's consider the function y = f(x), where f(x) represents the curve. We know that the point (a, b) lies on the graph of this function. Now, the slope of the tangent line at any point on the graph is equal to the derivative of the function evaluated at that point.

So, to find the slope of the tangent line, we need to find dy/dx (the derivative of y with respect to x) and evaluate it at x = a.

In this problem, the point on the graph is (a, 1/a). Therefore, the equation of the curve is y = f(x) = 1/x.

To find dy/dx, we differentiate the equation y = 1/x with respect to x using the power rule:

(d/dx) (1/x) = -1/x^2

Now, substituting x = a, we get:

dy/dx (at x = a) = -1/a^2

So, the slope of the tangent line is -1/a^2.

Regarding the confusion about the area of the triangle, let's break it down step by step:

The triangle is formed by the tangent line and the coordinate axes. The tangent line intersects the y-axis at the point (0, 2/a) and the x-axis at the point (-2a, 0).

The length of the base of the triangle is the distance between the x-intercept (-2a, 0) and the y-intercept (0, 2/a), which is 2a units.

The height of the triangle is the distance between the y-intercept (0, 2/a) and the point where the tangent passes through the curve (a, 1/a). This distance is 2/a units.

Now, the formula to calculate the area of a triangle is (base * height) / 2.

Using the base and height values we found, the area of the triangle is:

Area = (base * height) / 2
= (2a * 2/a) / 2
= 4/2
= 2

So, the area of the triangle formed by the tangent line and the coordinate axes is indeed 2.