Definition of Gradient Flashcards

All 6 cards in this deck

What does $\frac{dy}{dx}$ tell you about a curve?

It is the gradient function: it gives the gradient of the curve at any point, i.e. the rate of change of $y$ with respect to $x$.

The gradient of a curve at a particular point is equal to the gradient of what?

The gradient of the tangent to the curve at that point.

What are the steps to find the gradient of a curve at a given $x$-value?

e.g. find the gradient of $y = x^3 + 2x$ at $x = 2$

1. Differentiate: $\frac{dy}{dx} = 3x^2 + 2$
2. Substitute the $x$-value: $3 \times 2^2 + 2$
3. Gradient $= 14$

What are the steps to find the $x$-value at which a curve has a stated gradient?

e.g. find $x$ where $y = x^2 + 5x$ has gradient $11$

1. Differentiate: $\frac{dy}{dx} = 2x + 5$
2. Set the gradient function equal to the given value: $2x + 5 = 11$
3. Solve: $x = 3$

What does $\frac{d^2y}{dx^2}$ measure?

The rate of change of the gradient function — it is the derivative of $\frac{dy}{dx}$.

True or false? To find the $x$-values where a curve has zero gradient, you solve $\frac{dy}{dx} = 0$.

True. The gradient function is set to zero and solved as an equation.
e.g. $y = x^2 - 6x$ gives $2x - 6 = 0$.