Answered

Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.


Find the gradient of each of these functions at the point indicated.
a) y = x^2 at (3,9)
b) y = - 4x^3 at (1,-4)
c) y = 2x^4 at (-1,2)
d) y = 6x at (2, 12)

Sagot :

facundo3141592

Answer:

Remember that the gradient is equivalent o the derivative evaluated in one point.

And remember that if we have a function like:

f(x)  a*x^n

The derivative is:

f'(x) = n*a*x^(n - 1)

Then knowing this, let's start:

a) y = x^2

the derivative is:

y' = 2*x

And we want to evaluate this in the point (3, 9), then we just need to replace x by 3.

y'(3) = 2*3 = 6

The gradient of y = x^2 at (3, 9) is 6.

b) y = -4*x^3

The derivative is:

y' = 3*(-4*x^2) = -12*x^2

And we want to evaluate this on (1, -4), then we need to replace x by 1.

y'(1) = -12*1^2 = -12

The gradient of y = -4*x^3 at (1, -4) is -12.

c) y = 2*x^4

The derivative is:

y' = 4*2*(x^3) = 8*x^3

And we want to evaluate this on (-1, 2), then we need to replace x by -1.

y'(-1) = 8*(-1)^3 = -8

The gradient of y = 2*x^4 at the point (-1, 2) is -8

d) y = 6*x

The derivative is:

y' = 6

We can see that here we do not have any dependence on x, so the gradient will be the same for all values of x, this means that the gradient at the point (2, 12) is 6.

Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.