Index FAQ Introduction to Differentiation Tangents as Limits of Secant Lines Tangent Lines Linear Approximations of Functions Velocity Rates of Change

Index FAQ Tangents as Limits of Secant Lines The basic problem that leads to differentiation is to compute the slope of a tangent line of the graph of a given function f at a given point x 0. The key observation, which allows one to compute slopes of tangent lines is that the tangent is a certain limit of secant lines as illustrated in the picture below. x0x0 x 0 +h f A secant line intersects the graph of a function f at two or more points. The figure on the left shows secant lines intersecting the graph at the points corresponding to x=x 0 and x=x 0 + h. As h approaches 0, the secant line in question approaches the tangent line at the point (x 0,f(x 0 )).

Index FAQ Slopes of Secant Lines The slope of a secant line intersecting the graph of a function f at points corresponding to x=x 0 and x=x 0 + h can readily be computed using the notations defined in the picture below. x0x0 x0+hx0+h f h f(x 0 +h) f(x 0 ) f(x 0 +h)- f(x 0 ) As h approaches 0 (through positive numbers), the secant in the pictures approaches the tangent to the graph of f at the point (x 0,f(x 0 )).

Index FAQ 1 1 Tangent Lines (1) Definition Example Compute the slope of the tangent line, at the point (1,1), of the graph of the function x 2.

Index FAQ Tangent Lines (2) Solution Example Compute the slope of the tangent line, at the point (1,1), of the graph of the function x 2. Conclusion Equation of the tangent line is y-1=2(x-1), i.e., y=2x

Index FAQ Linear Approximations of Functions The following pictures show, in different scales, the graph of the function x 2 and that of its tangent line at the point (1,1). 0.77<x< <x<1.5 -1<x< 2 0.9<x<1.1 Conclusion Near the point of tangency, the tangent line approximates well the graph of the function. The closer we are the point of tangency, the better the approximation is.

Index FAQ The equation … … … (3..1) defining the derivative at c is equivalent to the equation, … … …(3.2) where the remainder rc(u) is of smaller order than u as, that is … … … (3.3) i.e. Differentiation and Linearization

Index FAQ Differentiation is Nothing but Local Linearization of Nonlinear Functions Affine LinearFunction

Index FAQ Velocity (1) Let f(t) denote the distance, in kilometers, a train has traveled in time t, t>0. Let h>0. The distance the train has traveled in the time interval from time t=t 0 to time t=t 0 +h is f(t 0 +h)-f(t 0 ). Hence thee average speed during this time interval is (f(t 0 +h)-f(t 0 ))/h. Taking the limit as h approaches 0 gives the speed of the train at time t=t 0. Problem Estimate the speed of the train at time t=t 0. Solution Conclusion

Index FAQ Velocity (2) Galileo made experiments that lead to the discovery of gravity. In the experiments he let various objects fall from the tower of Pisa. The top floor of the tower (above which the bells are hanging and from which objects can be dropped) is about 48 meters above the ground. Problem Given that the equation of motion for a freely falling object is s=f(t)=4.9t 2, compute the speed at which a freely falling object hits the ground when it is dropped from the top floor of the tower of Pisa.

Index FAQ Velocity (3) Problem Given that the equation of motion for a freely falling object is s=f(t)=4.9t 2, compute the speed at which a freely falling object hits the ground when it is dropped from the top floor of the tower of Pisa. Solution Let us first compute the speed of the object at time t=t 0. By the previous considerations we get: Conclusion

Index FAQ Velocity (4) Problem Given that the equation of motion for a freely falling object is s=f(t)=4.9t 2, compute the speed at which a freely falling object hits the ground when it is dropped from the top floor of the tower of Pisa. Solution (cont’d) Height of the towerDistance fallen in time t Conclusion

Index FAQ Rates of Change Definition

Index FAQ Applications of Rates of Change Depending on the situation, the rates of change of functions may model, for example, one of the following: 1.The slope of the tangent line. 2.The speed of an object. 3.Calculation of pdf from cdf 4.The rate at which an investment in a bank account grows. 5.The speed at which a hot object cools down or the speed at which a cold object warms up. 6.Population growth or decay.

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