Tangent to a Circle: A line is tangent to a circle if at a point if passes through and is perpendicular to the radius at . What does it mean to be tangent to a curve? We can generalize that it means one of the following: The tangent line, L, touches the curve, C, at only one place P, for all points on the curve “near P”. Below are pictures of two lines that are tangent to the curve and one line that is not tangent. Give an example of a physical interpretation of the tangent line to a curve: (car driving along curve, rock thrown from a sling) Discussion: Suppose we have a curve and we want to find the tangent to the curve at the point . We can construct secant lines joined by and points near . We choose points that approach . If the resulting slopes of the secant lines have a limit, say , we take it to be the slope of the curve at that point. The tangent line is the line that goes through and has slope . Ex 1 Find the derivative of at . What can we conclude? (Easiest derivatives to find.)
Tangent to a Circle: A line is tangent to a circle if at a point if passes through and is perpendicular to the radius at . What does it mean to be tangent to a curve? We can generalize that it means one of the following: The tangent line, L, touches the curve, C, at only one place P, for all points on the curve “near P”. Below are pictures of two lines that are tangent to the curve and one line that is not tangent. Give an example of a physical interpretation of the tangent line to a curve: (car driving along curve, rock thrown from a sling) Discussion: Suppose we have a curve and we want to find the tangent to the curve at the point . We can construct secant lines joined by and points near . We choose points that approach . If the resulting slopes of the secant lines have a limit, say , we take it to be the slope of the curve at that point. The tangent line is the line that goes through and has slope . Ex 1 Find the derivative of at . What can we conclude? (Easiest derivatives to find.)