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## 2.1 Tangents and Derivatives at a Point

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**Finding a Tangent to the Graph of a Function**To find a tangent to an arbitrary curve y=f(x) at a point P(x0,f(x0)), we • Calculate the slope of the secant through P and a nearby point Q(x0+h, f(x0+h)). • Then investigate the limit of the slope as h0.**Slope of the Curve**If the previous limit exists, we have the following definitions. Reminder: the equation of the tangent line to the curve at P is Y=f(x0)+m(x-x0) (point-slope equation)**Example**• Find the slope of the curve y=x2 at the point (2, 4)? • Then find an equation for the line tangent to the curve there. Solution**Rates of Change: Derivative at a Point**The expression is called the difference quotient of f at x0 with increment h. If the difference quotient has a limit as h approaches zero, that limit is named below.**2.2 The Derivative as a Function**We now investigate the derivative as a function derived from f by Considering the limit at each point x in the domain of f. If f’ exists at a particular x, we say that f is differentiable (has a derivative) at x. If f’ exists at every point in the domain of f, we call f is differentiable.**Alternative Formula for the Derivative**An equivalent definition of the derivative is as follows. (let z = x+h)**Calculating Derivatives from the Definition**The process of calculating a derivative is called differentiation. It can be denoted by Example. Differentiate Example. Differentiate for x>0.**Notations**There are many ways to denote the derivative of a function y = f(x). Some common alternative notations for the derivative are To indicate the value of a derivative at a specified number x=a, we use the notation**Graphing the Derivative**Given a graph y=f(x), we can plot the derivative of y=f(x) by estimating the slopes on the graph of f. That is, we plot the points (x, f’(x)) in the xy-plane and connect them with a smooth curve, which represents y=f’(x). Example: Graph the derivative of the function y=f(x) in the figure below.**What we can learn from the**graph of y=f’(x)?**Differentiable on an Interval; One-Sided Derivatives**If a function f is differentiable on an open interval (finite or infinite) if it has a derivative at each point of the interval. It is differentiable on a closed interval [a, b] if it is differentiable on the interior (a, b) and if the limits exist at the endpoints. Right-hand derivative at a Left-hand derivative at b A function has a derivative at a point if and only if the left-hand and right-hand derivatives there, and these one-sided derivatives are equal.**When Does A Function Not Have a Derivative at a Point**A function can fail to have a derivative at a point for several reasons, such as at points where the graph has • a corner, where the one-sided derivatives differ. • a cusp, where the slope of PQ approaches from one side and - from the other. • a vertical tangent, where the slope of PQ approaches from both sides or approaches - from both sides. • a discontinuity.**Differentiable Functions Are Continuous**Note: The converse of Theorem 1 is false. A function need not have a derivative at a point where it is continuous. For example, y=|x| is continuous at everywhere but is not differentiable at x=0.**Examples**Example.**Constant Multiple Rule**Note: Example.**Derivative Sum Rule**Example.**Derivative Product Rule**In function notation:**Example**Example: Solution:**Derivative Quotient Rule**In function notation:**Example**Example: Solution:**Second- and Higher-Order Derivatives**The derivative f’ of a function f is itself a function and hence may have a derivative of its own. If f’ is differentiable, then its derivative is denoted by f’’. So f’’=(f’)’ and is called the second derivative of f. Similarly, we have third, fourth, fifth, and even higher derivatives of f.**2.4 The Derivative as a Rate of Change**Thus, instantaneous rates are limits of average rates. When we say rate of change, we mean instantaneous rate of change.**Motion Along a Line: Displacement, Velocity, Speed,**Acceleration, and Jerk Suppose that an object is moving along a s-axis so that we know its position s on that line as a function of time t: s=f(t). The displacement of the object over the time interval from t to t+∆t is ∆s = f(t+ ∆t)-f(t); The average velocity of the object over that time interval is**Velocity**To find the body’s velocity at the exact instant t, we take the limit of the Average velocity over the interval from t to t+ ∆t as ∆t shrinks to zero. The limit is the derivative of f with respect to t.**Besides telling how fast an object is moving, its velocity**tells the direction of Motion.**The speedometer always shows speed, which is the absolute**value of velocity. Speed measures the rate of progress regardless of direction**The figure blow shows the velocity v=f’(t) of a particle**moving on a coordinate line., what can you say about the movement ?**Acceleration**The rate at which a body’s velocity changes is the body’s acceleration. The acceleration measures how quickly the body picks up or loses speed. A sudden change in acceleration is called a jerk.**Example**Near the surface of the earth all bodies fall with the same constant acceleration. In fact, we have s=(1/2)gt2 , where s is the distance fallen and g is the acceleration due to Earth’s gravity. With t in seconds, the value of g at sea lever is 32 ft/ sec2 or 9.8m/sec2.**Example**• Example: Figure left shows the free fall of a heavy ball bearing released from rest at time t=0. • How many meters does the ball fall in the first 2 sec? • What is its velocity, speed, and acceleration when t=2?**Example**Example: Solution:**Example**Example: Solution:**Example: A body hanging from a spring is stretched down 5**units beyond Its rest position and released at time t=0 to bob up and down. Its position at any later time is s=5cos t. What are its velocity and acceleration at time t?**Since**We have**Example**Example: Solution:**2.6 Exponential Functions**In general, if a1 is a positive constant, the function f(x)=ax is the exponential function with base a.**If x=n is a positive integer, then an=a a … a.**If x=0, then a0=1, If x=-n for some positive integer n, then If x=1/n for some positive integer n, then If x=p/q is any rational number, then If x is an irrational number, then**The Natural Exponential Function ex**The most important exponential function used for modeling natural, physical, and economic phenomena is the natural exponential function, whose base is a special number e. The number e is irrational, and its value is 2.718281828 to nine decimal places.**Derivative of the Natural Exponential Function**Example. Find the derivative of y=e-x. Solution: Example. Find the derivative of y=e-1/x.