Equation of a tangent Instantaneous rate of change

Equation of a tangent. • Instantaneous rate of change is given by the gradient of the tangent to the given point on a curve. • Draw a tangent, pick up two points, estimate the gradient of the tangent. • In Calculus we can determine the exact equation of the tangent to a curve at a given point. 1 of 52

Gradient & Intercept Review Definition The general equation of a straight line can be written as: y = mx + c The value of m tells us the gradient of the line. The value of c tells us where the line crosses the y-axis. This is called the y-intercept and it has the coordinate (0, c). For example, the line y = 3 x + 4 has a gradient of 3 and crosses the yaxis at the point (0, 4). 2 of 52

Equation of tangent using calculus Problem: Find the equation of the tangent to the curve y=x 2+2 at a point where x=1. To find the gradient, m, follow these steps: 1. Find dy/dx 2. Find the gradient at x=1 3. Write the equation as y=mx+c, we know the gradient now. 4. Find the y-value of the point. 5. Substitute the point into the equation to find c. 3 of 52

Problem: Find the equation of the tangent to the curve y=x 2+2 at a point where x=1. Therefore m=2 and y=2 x+c When x=1, y=12+2=3, so (1, 3) lies on the parabola. 3=2(1)+c, c=1 Equation of the tangent is y=2 x+1 4 of 52

Check with your GDC: Draw the graph 5 of 52 Find the gradient at x=1 Draw a tangent at this point

Your turn Question 1: Find the equation of the tangent to the curve y=2 -x 3 at a point x=-2. 6 of 52

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