This gives us the secant line equation y = 4 x + b. For part b, we’ll find the slope of the line using the derivative of f ( x) at x = 1 or f ′(1).įor part a, the line must pass through the point (1, 2) and (5,18) since For part a, we’ll find the slope of the line using two points on the function. With this in mind, you’ll have no trouble tackling tangent line problems on the AP Calculus exam!įor more about slope, tangent lines, and derivatives, check out these related Magoosh articles: Is the Derivative of a Function the Slope? and How to Find the Slope of a Line Tangent to a Curve.Solution For each of these parts, we’ll find the equation of a line using y = mx + b. Instead, the derivative is a tool for measuring the slope of the tangent line at any particular point, just like a clock measures times throughout the day. The derivative is not the same thing as a tangent line. Now let’s see the graph of y = f( x) together with the tangent lines that we just found. So I’ve shown you both the point-slope form and the simplified, or slope-intercept, form of the tangent line. Note, on the AP Calculus Exam, the multiple choice answers may be simplified. The function value and derivative value at a few points are shown in the table below:įor the points listed, we can easily find the equation of the tangent line. (for practice finding derivatives, check out this Magoosh article about derivatives). We’ll find the tangent lines at a few different points.įirst of all, find the derivative: f ‘( x) = 3 x 2 + 6 x. Now let’s look at an example function, f( x) = x 3 + 3 x 2 + 1. And if I want to know the slope at x = -352/13, then I compute f ‘(-352/13). If I want to know the slope of f at x = 1, then I compute f ‘(1). This is what we mean when we say that “the derivative measures the slope of the tangent lines.” By plugging in different input values, x = a, the output values of f ‘( x) give you the slopes of the tangent lines at each point x = a. Let’s take another look at that first step, “ Find the derivative.” Remember, the derivative is a function (of the input variable x). In other words, plug in your values of m, a, and b into the equation, Use the point-slope form and solve for y to find the equation of the tangent line.As always, you plug the x-value into the function in order to get the y-value. If not already given in the problem, find the y-coordinate of the point.Suppose you are asked to find the tangent line for a function f( x) at a given point x = a. Once you have the slope, writing the equation of the tangent line is fairly straightforward. Your job is to find m, which represents the slope of the tangent line. In this formula, the function f and x-value a are given. So how do we know what the slope of the tangent line should be? After learning about derivatives, you get to use the simple formula, The graph below shows the tangent lines (in red, purple, and magenta) at three different points on a curve y = f( x) (in black). Sometimes we might say that a tangent line “ just touches” the curve, or “ intersects the curve only once,”f but those ideas can sometimes lead us astray. What is a tangent Line?Ī tangent line for a function f( x) at a given point x = a is a line (linear function) that meets the graph of the function at x = a and has the same slope as the curve does at that point. Let’s take a closer look at tangent lines. A clock measures the time at any particular point throughout the day. But if you want to know the time of day, you can go look at a clock to find out. Think about this: a clock is not the same thing as time. Instead, the correct statement is this: “The derivative measures the slope of the tangent lines.” For starters, the derivative f ‘( x) is a function, while the tangent line is, well, a line. So there’s a close relationship between derivatives and tangent lines. In calculus, we learn that the tangent line for a function can be found by computing the derivative. By Shaun Ault on Janu, UPDATED ON June 14, 2022, in AP
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