(MTH 250) Lecture 8 Calculus. Previous Lecture’s Summary Formal definition of continuity Important results of continuity Hyperbolic trigonometric functions.

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Presentation transcript:

(MTH 250) Lecture 8 Calculus

Previous Lecture’s Summary Formal definition of continuity Important results of continuity Hyperbolic trigonometric functions Inverse hyperbolic trigonometric functions Boundedness of continuous functions Uniform continuity Tangent lines

Today’s Lecture Tangent lines: Recalls Introduction to derivatives Derivatives as rate of change Differentiability and continuity Non-differentiable functions Some useful derivatives

Tangent lines: Recalls

Common Misconceptions: A line is tangent to a curve if it touches the curve only once. (wrong) A line is tangent to a curve if it crosses the curve in exactly one point. (wrong) A line is tangent to a curve if it touches the curve at only one point but does not cross the curve. (wrong) Tangent lines: Recalls Tangent line Not Tangent line

Tangent lines: Recalls

Introduction to derivative

Derivative as rate of change One of the most important themes in calculus is the study of motion. To describe the motion of an object completely, one must specify its speed and the direction in which it is moving. The speed and direction together comprise what is called velocity of the object. For the moment, we will consider only the motion of an object along a line: this is called rectilinear motion. Examples: A car moving along a straight track, motion of a free falling body, ball thrown straight up and coming down along the same path.

Derivative as rate of change

Derivative as rate of change

Differentiability and continuity

Cusp: A cusp is a point on the graph at which the function is continuous but the derivative is discontinuous. Verbally, cusp is a sharp point or an abrupt change in direction. Non-differentiable functions

If a function is discontinuous at a point then it is not differentiable at that point. As the slope of a vertical line is undefined. Therefore, if the slope of the tangent line to a graph is vertical at a point, then the derivative of the function will be undefined at that point. It is possible for a function to be continuous at a point but not be differentiable i.e. the function has a cusp at some point, then the function is not differentiable at that point. If the graph has a cusp at a point, it would be possible to draw an infinite number of lines that look like tangent lines, and so there is finally no tangent line. Non-differentiable functions

Not Differentiable Not Continuous Still Continuous Non-differentiable functions

Useful derivatives

Lecture Summary Tangent lines: Recalls Introduction to derivatives Derivatives as rate of change Differentiability and continuity Non-differentiable functions Some useful derivatives