The Derivative And The Tangent Line Problem

1/21/2014The Derivative and the Tangent LineProblemCalculus grew out of four major problems that Europeanmathematicians were working on during the seventeenth century.1. The tangent line problem2. The velocity and acceleration problem3. The minimum and maximum problem4. The area problemEach problem involves the notion of a limit, and calculus can beintroduced with any of the four problems.The Tangent Line Problem What does it mean to say that a line is tangentto a curve at a point? For a circle, the tangentline at a point P is the line that is perpendicularto the radial line at point P, as shown in Figure3.1.Tangent line to a circleFigure 3.11

1/21/2014The Tangent Line ProblemFor a general curve, however, the problem is more difficult.For example, how would you define the tangent linesshown in Figure 3.2?Tangent line to a curve at a pointFigure 3.2The Tangent Line ProblemYou might say that a line is tangent to a curve at a point Pif it touches, but does not cross, the curve at point P.This definition would work for the first curve shown inFigure 3.2, but not for the second.Or you might say that a line is tangent to a curve if the linetouches or intersects the curve at exactly one point.This definition would work for a circle but not for moregeneral curves, as the third curve in Figure 3.2 shows.2

1/21/2014The Tangent Line ProblemEssentially, the problem of finding the tangent line at apoint P boils down to the problem of finding the slope ofthe tangent line at point P.You can approximate this slopeusing a secant line through thepoint of tangency and a secondpoint on the curve, as shown inFigure 3.3.The secant line through (c, f(c))and (c x, f(c x))Figure 3.3The Tangent Line ProblemIf (c, f(c)) is the point of tangency and (c x, f(c x)) is asecond point on the graph of f, the slope of the secant linethrough the two points is given by substitution into theslope formulaSlope of secant lineThe right-hand side of this equation is a differencequotient.3

1/21/2014The Tangent Line ProblemThe denominator x is the change in x, and the numerator y f(c x) – f(c) is the change in y.The Tangent Line ProblemThe slope of the tangent line to the graph of f at thepoint (c, f(c)) is also called the slope of the graph of f atx c.4

1/21/2014Example 1 – The Slope of the Graph of a Linear FunctionFind the slope of the graph of f(x) 2x – 3 at the point (2, 1).Solution:To find the slope of the graph of f when c 2, you canapply the definition of the slope of a tangent line. 2The Tangent Line ProblemThe graph of a linear function has the same slope at anypoint. This is not true of nonlinear functions.The definition of a tangent line to a curve does not coverthe possibility of a vertical tangent line. For vertical tangentlines, you can use the following definition.If f is continuous at c andthe vertical line x c passing through (c, f(c)) is a verticaltangent line to the graph of f.5

1/21/2014The Tangent Line ProblemFor example, the function shown in Figure 3.7 has avertical tangent line at (c, f(c)).If the domain of f is the closedinterval [a, b], you can extendthe definition of a verticaltangent line to include theendpoints by consideringcontinuity and limits from theright (for x a) and from theleft (for x b).The graph of f has a verticaltangent line at (c, f(c)).Figure 3.7The Derivative of a FunctionYou have now arrived at a crucial point in the study ofcalculus. The limit used to define the slope of a tangent lineis also used to define one of the two fundamentaloperations of calculus—differentiation.6

1/21/2014The Derivative of a FunctionBe sure you see that the derivative of a function of x is alsoa function of x.This “new” function gives the slope of the tangent line tothe graph of f at the point (x, f(x)), provided that the graphhas a tangent line at this point.The process of finding the derivative of a function is calleddifferentiation.A function is differentiable at x if its derivative exists at xand is differentiable on an open interval (a, b) if it isdifferentiable at every point in the interval.The Derivative of a FunctionIn addition to f ′(x), which is read as “f prime of x,” othernotations are used to denote the derivative of y f(x).The most common areNotation for derivativesThe notation dy/dx is read as “the derivative of y withrespect to x” or simply “dy, dx.”7

1/21/2014The Derivative of a FunctionUsing limit notation, you can writeExample 3 – Finding the Derivative by the Limit ProcessFind the derivative of f(x) x3 2x.Definition of derivative8

1/21/2014The Derivative of a FunctionRemember that the derivative of a function f is itself afunction, which can be used to find the slope of the tangentline at the point (x, f(x)) on the graph of f.Find the slope of the tangent line to the graph of thefunction at the given point.g ( x) 6 x 2 , (1,5)9

1/21/2014Differentiability and ContinuityThe following alternative limit form of the derivative isuseful in investigating the relationship betweendifferentiability and continuity. The derivative of f at c isAlternative form of derivativeprovided this limit exists(see Figure 3.10).As x approaches c, the secant lineapproaches the tangent line.Figure 3.10Differentiability and ContinuityNote that the existence of the limit in this alternative formrequires that the one-sided limitsexist and are equal. These one-sided limits are called thederivatives from the left and from the right, respectively.It follows that f is differentiable on the closedinterval [a, b] if it is differentiable on (a, b) and if thederivative from the right at a and the derivative from the leftat b both exist.10

1/21/2014Differentiability and ContinuityIf a function is not continuous at x c, it is also notdifferentiable at x c. For instance, the greatest integerfunctionis not continuous at x 0, andso it is not differentiable at x 0,(see Figure 3.11).The greatest integer function is not differentiableat x 0, because it is not continuous at x 0.Figure 3.11Differentiability and ContinuityAlthough it is true that differentiability implies continuity (asin Theorem 3.1), the converse is not true.That is, it is possible for a function to be continuous at x cand not differentiable at x c.11

1/21/2014Example – A Graph with a Sharp TurnThe functionf(x) x – 2 shown in Figure 3.12 is continuous at x 2.f is not differentiable at x 2, because thederivatives from the left and from the rightare not equal.Figure 3.12Example – A Graph with a Sharp Turncont’dHowever, the one-sided limitsDerivative from the left –1andDerivative from the right 1are not equal.12

1/21/2014Example – A Graph with a Vertical Tangentf is not differentiable at x 0,because f has a vertical tangent line at x 0.Differentiability and ContinuityYou can summarize the relationship between continuityand differentiability as follows.1. If a function is differentiable at x c, then it is continuousat x c. So, differentiability implies continuity.2. It is possible for a function to be continuous at x c andnot be differentiable at x c. So, continuity does notimply differentiability.13

1/21/2014A function is not differentiable at x c if:1) the function is discontinuous at c2) the graph of the function comes to a point at c whichmeans the derivatives from the left and from theright are not equal3) the function has a vertical tangent line at x cFind an equation of the tangent line to the graph of fat the given point.f ( x) x 3 1, (1, 2)We are looking for the equation of a line of which we know a point.Now we need the slope so we can use the point-slope form of theequation of a line.y y1 m( x x1 )14

1/21/2014To find the slope, we must find the derivative.f ( x) x3 1f ' ( x) lim x 0f ( x x) f ( x) x ( x x )3 1 ( x3 1) lim x 0 x15

1/21/2014 1 The Derivative and the Tangent Line Problem Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century. 1. The tangent line problem 2. The velocity and acceleration probl