Detailed Solutions to Initial Practice
Unit 3: Determining the Slope of a Curve at the Point of Tangency

1. Which have a line drawn tangent to the curve?
 Figure (a): The straight line is not a tangent line. It crosses the curve at two points, A and B. Figure (b): The straight line is tangent to the curve. The straight line touches the curve at point A, but does not cross the line. Figure (c): The line is not a tangent line. It does touch the curve at a single point, D, but it also crosses the curve at this point. Figure (d): The straight line is tangent to the curve. The straight line touches the curve at point C but does not cross the line.

2. Find the slope of the following curve at point E.

The straight line JK is tangent to the curve at point E. (Note that line RT is not tangent to the curve but intersects it at point E.) To calculate the slope of the curve at point E, you need to calculate the slope of line JK.

 Step One: Identify two points on the line. Identify points J (0, 2) and K (4, 6) on the line. Step Two: Select one to be (x1, y1) and the other to be (x2, y2). Let point J (0, 2) to be (x1, y1). Let point K (4, 6) to be (x2, y2). Step Three: Use the slope equation to calculate slope. Using points J (0, 2) and K (4, 6), your calculations will look like: The slope of the curve at point E is 1.

If you had problems with this question, please reread the section of this unit on calculating the slope at a point on a curve. If you feel you need further review on calculating the slope of a straight line, you should review the unit on Calculating Slope. Then return to this unit and try the additional practice.