Math 1410 Worksheet #27: Section 4.9 Name: Our final application of derivatives is a prelude to what will come in later chapters. In many situations, it will be necessary to find a way to reverse the differentiation process. For example, an individual could know the velocity of an object (rate of change of a quantity) and want to find its position at some time (the actual quantity at a given state). Since the derivative of position is velocity, the individual is looking to reverse the differentiation process. In general, if we start with a function f and want to reverse the differentiation process, then we are finding an antiderivative of f. Definition 1. A function g is an antiderivative (AD) of f on an interval if Problem 1. Find an antiderivative of f(x) = x 27. Notice in the previous example that we can come up with many, many antiderivatives, each of which differ only by the constant term. So, is there a way of finding all of the antiderivatives of a function f? Yes! The following theorem tells us how to do this: Theorem 1. If g is any antiderivative of f on an interval, then the most general antiderivative of f is This theorem is a consequence of two theorems from Section 4.2, which we ll list here: Theorem 2. If f (x) = 0 for all x in (a, b), then f is constant on (a, b). Theorem 3. If f (x) = g (x) for all x in (a, b), then f(x) = g(x) + c for some constant c. In other words, f and g differ only by a vertical shift.
Problem 2. Let s practice finding antiderivatives. following functions. Find the most general antiderivative of the (a) f(x) = x n, where n 1 (b) f(x) = sin(x) (c) f(x) = cos(x) (d) f(x) = sec 2 (x) (e) f(x) = csc(x) cot(x)
(f) f(x) = 3 (g) f(x) = 8x 4 + 12x 9 3 x 7 (h) f(x) = sin(16x) + πx 2 3 x 145 (i) f(x) = 6x 4/5 csc 2 (x) The table on page 276 gives a partial list of antiderivatives. You should look at this table! Equations containing derivatives of a function are called differential equations. Differential equations represent a whole field of study and active research. In this class, we will only look at basic types where we are given a derivative of some sort and asked to find the original function. This is what we have done in the above problems.
In some cases, additional information is given so that we can determine a value for C for that situation. This is quite necessary for various physical applications. Problem 3. Compute f if f (x) = sin(x) x + 11π and f(0) = 6. Problem 4. Determine f(x) if f (x) = 12x 5 + sin(x) + 9, f(0) = 7, f (0) = 3.
We have noticed in earlier work that The antiderivative of f (x) + g (x) is f(x) + g(x) + C. The antiderivative of f (x) g (x) is f(x) g(x) + C. WARNING!!!!!!! The antiderivative of f (x)g (x) is NOT f(x)g(x) + C. The antiderivative of f (x)/g (x) is NOT f(x)/g(x) + C. These cases require more advanced techniques to find the antiderivative and will be covered later in this class or in Calculus II. Antidifferentiation can be used in the context of position, velocity, and acceleration. Recall that the derivative of position is velocity, and the derivative of velocity is acceleration. So... The antiderivative of acceleration is. The antiderivative of velocity is. Problem 5. Suppose I stand on the top of a cliff that is 100 m high. I hold a ball over the cliff s edge and throw it upward with a speed of 25 m/s. (a) Given an equation for the height of the ball above the ground t seconds after I throw it. (b) When does the ball reach its maximum height?
Back in Section 3.2, we developed techniques for drawing a graph of f if we had a graph of f. Keep in mind that the slopes of the tangent lines to the graph of f are the y-coordinates of the graph of f. Now we will reverse the process. Suppose we have a graph of f. If we want to draw the graph of f, then we are drawing the antiderivative of f. This can be done with the following steps. If the graph of f is positive (above the x-axis), then the slope of the tangent line to f is If the graph of f is negative (below the x-axis), then the slope of the tangent line to f is. If the graph of f is increasing, then.. This means the graph of f is. If the graph of f is decreasing, then. This means the graph of f is. If lim x f (x) = 0, then the graph of f. The results are the same if x. If lim x f (x) = c, then the graph of f. The results are the same if x.
Problem 6. Below is a graph of f (x). Use this to draw a rough sketch of f(x) if we know f(0) = 1.