Slope of a Curve: Tangent Lines and Instantaneous Rates of Change, Lecture notes of Calculus

Download Slope of a Curve: Tangent Lines and Instantaneous Rates of Change and more Lecture notes Calculus in PDF only on Docsity! MATH 1142 Section 1.2 & 1.3 Worksheet NAME Slope of a Curve at a Point We want to extend the notion of of a slope of a line to the slope of more general curves. However, this is somewhat problematic. Why? One way to think of extending the notion of the slope of a line is to look for the slope of the line tangent to the curve at a particular point. Slope of a Curve The slope of a curve at a point P is defined to be the slope of the tangent line to the curve at P . What does this mean? The slope of a curve at a point P measures the instantaneous rate of change measures the rate of change of the curve as it passes through P . Will the slope of a curve be the same for any point on the curve? Why or why not? What does this mean about the slope of a curve? For most functions, there is a formula for finding the slope of a curve, f(x), this formula is called the called the derivative (or sometimes the slope formula) and is denoted f ′(x). Recall that we already know the slope of a line g(x) = mx+ b is this means that the derivative of the line is g′(x) = . Calculating this derivative can be more complicated when dealing with more complicated curves since we are trying to find the slope of a line for which we only know one point. One way to approximate the derivative is by calculating the slope of the secant line as opposed to the tangent line (since we will know two points). Calculating the slope of the secant line is the same as finding the average rate of change of the function between two points. 1 How would you calculate the rate of change of a function f(x) between the points x = a and x = b? Sketch a picture/graph that describes what is happening in the previous formula. If f were a linear function, this value would calculate the slope of that line. Since f may not be a linear function, we say that this value is calculating the slope of the secant line through (a, f(a)) and (b, f(b)) of f . However, what we want to find is the slope of the tangent line at one particular point (a, f(a)). This will give us the instantaneous rate of change. Definition The tangent line to the curve y = f(x) at the point (a, f(a)) is the line through (a, f(a)) with slope f ′(a) (provided that this limit exists). If f ′(a) is the slope of the tangent line at a, what is an equation of the tangent line at x = a? Equation for the Tangent Line Explain why f ′(a) models that instantaneous rate of change of f at (a, f(a)). (You may want to use your graph/picture.) 2