Finding the equations of the tangents
A. Tangent at a Given Point on the Curve
Move the slider m until the tangent line is obtained. The slope of the tangent = =
The equation of the tangent at A(4, 1) in the form of y = mx + c is
B. Tangent to a Curve with a Given Slope
Move the slider of c to obtain tangents to the curve. How many tangents do you get?
a. Slope of the tangent by differentiation
Let A be (a, b). By considering the first derivative at A, express the slope of the tangent at A in terms of a and b.
b. Slope of the tangent by slope of L
By considering the slope of L, the slope of the tangent at A =
c. Find the point of contact A (a, b)
By combining the result of a. and b., find the value(s) of a.
The equation of the tangents at A is (you may choose more than one answer.)
C. Tangents to the Curve from an External Point
Move the slider of m to obtain a tangent to C. How many tangent(s) do you get?
a. Slope of the tangent by differentiation
Let the point of contact be (a, b). By considering the first derivative at A, express the slope of the tangent at A in terms of a and b.
b. Slope of the tangent by considering the given point.
By two-point form, the slope of the tangent at (a, b) =
Equation 1
Combining the result of a. and b., we obtain an equation in a and b as follow.
Equation 2
Considering the point (a, b) lies on the curve C, set up an equation for a and b.
Finding (a, b)
Solving Equation 1 and Equation 2, we obtain (a, b).
c. equation of the tangent
By using two point form, find the equations of the tangent at (a, b) to the curve.