Question

Consider the parabola y = 8x − x2. (a) find the slope of the tangent line to the parabola at the point (1, 7). 6 (b) find an equation of the tangent line in part (a)

Answer 1

Answer:

a) 6

b) $y = 6x + 1$

Step-by-step explanation:

Suppose we have a function y = f(x).

The slope of the tangent line at the point x = x0 is

$m = y'(x0)$

In this problem, we have that:

$y = 8x - x^{2}$

(a) find the slope of the tangent line to the parabola at the point (1, 7).

$x_{0} = 1$

So

$m =  y'(1) = 8 - 2x = 8 - 2 = 6$

(b) find an equation of the tangent line in part (a)

We have a point $(x_{0}, y_{0})$

The equation for the tangent line is:

$y - y_{0} = m(x - x_{0})$

So

$x_{0} = 1, y_{0} = 7$

$y - y_{0} = m(x - x_{0})$

$y - 7 = 6(x - 1)$

$y = 6x - 6 + 7$

$y = 6x + 1$

Answer 2

The slope of the tangent line to a point in any given function is equal to the derivate of the function evaluated in that point.

The solutions are:

• a) The slope is 6.
• b) y = 6*x + 1

Our function is y = f(x) = 8*x - x^2

a) Here we need to derivate our function, we will get:

f'(x) = 8 - 2*x

To get the slope of the tangent line at the point (1, 7) we need to evaluate the derivated function at the same x-value of the point, we will get:

x = 1

Then:

f'(1) = 8 - 2*1 = 8 - 2 = 6

The slope is 6.

b) We want to find the equation of the line.

Remember that a general line can be written as:

y = a*x + b

where a is the slope and b is the y-intercept.

Here we already know the slope, so we have a = 6.

Then our line becomes:

y = 6*x + b

To find the value of b we can use the fact that this line passes through the point (1, 7)

This means that when x = 1, we have y = 7.

Replacing these in the line equation we get:

7 = 6*1 + b

7 = 6 + b

7 - 6 = b

1 = b

Then the equation of the line is:

y = 6*x + 1

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