The tangent to the curve y=x^2+3 x will pass through the point (0,-9) if it is at the point (a)

step 1 Here we are given the curve y is equal to x squared plus 3x now the gradient expression to y d x

The tangent to the curve y=x^2+3 x will pass through the...

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Key Concepts

External Point Constraint

In many problems, a condition is set whereby the tangent line must pass through an external point that is not on the curve. This imposes an extra constraint which, when applied along with the point-slope form of the tangent line, allows one to solve for unknown parameters or determine the specific point of tangency on the curve.

Point-Slope Form

The point-slope form is an equation of a line used to describe a line given its slope and any point on the line. This form is particularly useful when determining the equation of the tangent line, as it directly incorporates the slope (obtained from the derivative) and the coordinates of the point of tangency.

Derivative

The derivative of a function gives the instantaneous rate of change at any point on its graph. It is used to compute the slope of the tangent line to the curve at a specific point, which is fundamental in calculus for understanding how functions behave locally.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve only at that point and has the same slope as the curve does at that point. It's a primary concept in differential calculus and is used to approximate the behavior of functions near the point of tangency.

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