Question

Please help with the questions in the image

Answer 1

Step-by-step explanation:

This seems to be calculus 1.

Question a

We have $-2x^3 + 5x^2 + 9$

m = slope = derivative

Find the derivative / slope of $-2x^3 + 5x^2 + 9$

We do this by differentiating the polynomials. There are a few methods to do this but I am going to use the power rule, which we multiply the constant by the exponent on the variable and subtract one from the exponent.

$\frac{d}{dx}(-2x^3 + 5x^2 + 9)$

$-6x^2 + 10x$

$m = -6x^2 + 10x$ when x = a

Now that we have this information, we can answer question b

Question b

The tangent line for Point (1, 12)

First find the slope by using our derivative.

$m = -6x^2 + 10x$

$m = -6(1)^2 + 10(1)$

$m = -6 + 10x$

$m = 4$

Now that we have our slope, use point slope form to find our tangent line

$y - 12 = 4(x - 1)$

$y(x) = 4x + 8$

Now lets do the same for the Point (2, 13)

Find the slope at the point.

$m = -6x^2 + 10x$

$m = -6(2)^2 + 10(2)$

$m = -24 + 10$

$m = -4$

Now find the tangent line using point slope form of a line.

$y(x) - 13 = -4(x - 2)$

$y(x) = -4x + 21$

Now graph the lines, which I have done and you can see by viewing the image I have attached.