Tangent Lines and Derivatives

Transcription

1 The Derivative and the Slope of a Graph Tangent Lines and Derivatives Recall that the slope of a line is sometimes referred to as a rate of change. In particular, we are referencing the rate at which the variable y changes with respect to the change in the variable x. This is a practical concept since there are many examples in real life where we wish to identify changes in one variable as a result of changes in another. Remember that the slope of a line was dependent on two points. In order to compute this rate of change we needed to know the change in two variables. This gave us the rate of change between two points on a curve. In calculus, we are interested in finding the rate of change at one point. We wish to find the slope of a tangent line to a curve. Tangent means to touch and so we are looking for the line that touches the curve at one point. Here are some examples. The tangent line (green line) in both of those examples is adjacent to the curve at one point. y2 y1 When we want to find the slope of a line, we use the formula m. When we wish to find the x2 x1 slope of the tangent line we encounter the problem that we do not have two points. A line that goes through two points on the graph is called the secant line. The slope of the secant line is as above, y2 y1 m, but we can express this in a manner that better serves us to find the tangent line. x x 2 1 We are interested in finding the slope at just one point, ( x, y). Remember that y is the same as the function value at x or y= f(x). So we will use the two points ( x, ) and ( x, f ( x )). The represents the change in x. It is the amount necessary to be added (or subtracted) from x to get to the x from the second point. Then we can find the slope of any secant line to be the following: m sec f ( x ) ( x ) x f ( x )

2 We say h.) f ( x ) is called the difference quotient. (Sometimes is replaced by the variable If we look at a graph where we are interested in the finding the slope of the tangent line, we can use a progression of secant lines to get closer and closer to the slope of the secant line. (You should be thinking limits at this point in time.) Consider the following graphs. Can you identify the secant lines? Where is the tangent line? What is happening to the secant lines in conjunction with the tangent line?

3 Definition of the Slope of a Graph The slope m of the graph of f at the point (x,f(x)) is equal to the slope of its tangent line at (x,f(x)) and is given by f ( x ) m lim msec lim 0 0 provided the limit exits. Definition: The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope f ( a) lim xa x a provided this limit exists. Examples: Find the slope of the graph of 2 x at the point 3,9) (. We find lim 0 f ( x )

4 Example: Find the equation for the slope of the tangent line to any point on the graph 3x 5. Then graphically interpret these results. Find the equation that represents the slope of the graph Or at the point (3, 39)? 5x 2 2x. What is the slope for x=5?

5 Definition of the Derivative The derivative of f at x is given by f ( x ) f '( x) lim provided the limit exists. 0 A function is differentiable at x if its derivative exists at x. The process of finding derivatives is called differentiation. There are other notations we use to describe the derivative: dy d f '( x) y' f ( x) Dxy dx dx It is also common to replace h giving us f '( x) lim h0 f ( x h) h. Example: Find the derivative of 1. x Example: d 4x 5 dx Find

6 Find d dx 8 d Find x dx The Velocity Problem The first derivative of the position function is the velocity function. We call this instantaneous velocity or velocity.

7 Differentiability and Continuity Consider a polynomial function, the Heaviside function, and the absolute value function. Do you note any places where the function is not differentiable? Recall the Heaviside function is 1 if x 0 H ( x). 1 if x 0 What connection do you make between differentiability and continuity? Differentiability Implies Continuity If a function f is differentiable at x=c then f is continuous at x=c.