Anna D Aloise May 2, 2017 INTD 302: Final Project Demonstrate an Understanding of the Fundamental Concepts of Calculus Analyzing the concept of limit numerically, algebraically, graphically, and in writing. Analyzing a limit numerically: In order to analyze the concept of a limit numerically, one must examine an x and y chart which contains the points of the function. By doing so, we are able to tell from the y values which x value the limit is approaching by identifying which value F (x) is getting infinitely close to from both sides. Consider the following table, which more clearly demonstrates this process: Estimate the lim (3x 2) numerically X 1.9 1.99 1.999 2.0 2.001 2.01 2.1 F (x) 3.700 3.970 3.997? 4.003 4.030 4.300 Using the table above, it is easy to estimate numerically that the values appear to approach 4 from the left and from the right. Analyzing a limit Algebraically: In order to analyze the concept of a limit algebraically, we are able to use direct substitution of the x value into our function. However, this method can only be done if the function is continuous. The best way to check for this is to substitute the x value into the function to see if the denominator is undefined. If it is, the function is not continuous and this method will not work and there exists a discontinuity somewhere within the function. Analyzing a limit Graphically: When analyzing a limit graphically, the first step is to identify any discontinuities within the graph. Possible tpes of discontinuities: 1. Removable Discontinuity: exists when the limit of the function exists, but one or both of the other two conditions is not met, this is more commonly referred to as whole. 2. Infinite Discontinuity: exists when one of the one-sided limits of the function is infinite. Graphically, this situation corresponds to a vertical asymptote. Many rational functions exhibit this type of behavior.
3. Finite Discontinuity: exists when the two-sided limit does not exist, but the two one-sided limits are both finite, yet not equal to each other. The graph of a function having this feature will show a vertical gap between the two branches of the function. This is commonly referred to as a jump discontinuity. 4. Oscillating Discontinuity: exists when the values of the function appear to be approaching two or more values simultaneously and is vacillating to much to have a limit as x 0. If the graph does not contain any of the discontinuities above, it is continuous at every point in its domain. Analyzing a limit through Writing: When analyzing a limit through writing, one must clearly explain what is happening to x as it approaches a given value from the left and the right using appropriate grade level terminology. When deciding which of the four methods to use in order to analyze the concept of a limit, it is preferred to do so algebraically since it is more precise and leaves less room for error. Interpreting the derivatives as the limit of the difference quotient. The difference quotient is the definition of the derivative: The derivative of f (x) with respect to x is the function f (x) and is defined as, f x = lim h 0 f x 0 + h f(x 0 ) h We can interpret this in the following ways: 1. The slope of the graph of y = f x at x = x 2. The slope of the tangent to the curve y = f x at x = x 3. The slope of a secant line through two points on the graph of f. 4. The rate of change of f (x) with respect to x at x = x 5. The derivative f (x ) at a point We use the definition of the derivative in the following ways: 1. To determine if f is differentiable at x, if it is, then f is continuous at x 2. To differentiate polynomials 3. To use the product and quotient rules to differentiate. Interpreting the definite integral as the limit of a Riemann sum. We use the limit of Riemann Sums when we are approximating the area under the curve by partitioning the area into a finite number of rectangles. The more partitions, rectangles, we create, the better the approximation of the area under the curve will be. Most commonly, the following formula is used:
n A = lim f x i x = f x dx n i1 a Where n is equal to the number of subdivisions, each with a width of x = () and x is the amount of choices of points for the partitions. The limit of any Riemann sum is taken as the norm of the partitions approaches zero and the number of subdivisions goes to infinity. When the condition in the definition is satisfied, we say the Riemann sum of f on [a, b] converges to the definite integral where the upper bound is b and the lower bound is a. Applying the Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus connects integration and differentiation to compute integrals using an antiderivative of the function. In addition, the rate of change of the area under the curve up to a specific point equals the height of the area at that point. In order to better understand this theorem, we will break it down into two parts: Part 1: The first part of the theorem discusses the relationship between the derivative and the integral. If f is continuous on [a, b] where b is the upper bound and a is the lower bound, then F x = f x dx is continuous on [a, b], differentiable on (a, b) and its derivative is f (x): F x = d dx b b f x dx = f(x) a Part 2: The second part of the theorem describes how to evaluate definite integrals without having to calculate limits of Riemann Sums. Here we find and evaluate an antiderivative at the upper and lower bounds of integration. If f is continuous at every point in [a, b] and F is any antiderivative of f on [a, b], then a b f x dx = F b F(a) Properties of the Fundamental Theorem of Calculus: f x dx = 0 f x dx = f x dx + f x dx Notice we are able to interchange the limits of integration b a f x dx = a b f x dx Applying the concepts of derivatives to interpret gradients, tangents, and slopes. Applying the concept of derivatives to interpret gradients: Computed at the vector of each of the partial derivatives. Gradient in its simplest form is a synonym for slope and is denoted by,. The gradient points in the direction of the greatest rate of increase of the function. The gradient vector of f x, y at a point P x, y is the vector
f = " " i + j Obtained by evaluating the partial derivatives of f at P " " Applying the concept of derivatives to interpret tangents: A tangent line is a straight line that touches a function at only one point. This line also represents the instantaneous rate of change of the function at one point. The slope of the tangent line at a point on the function is equal to the derivative of the function at the same point. Applying the concept of derivatives to interpret slopes: This is equivalent to the average rate of change, or the slope, between two given points. A secant line is a straight line jointing two points on a function. Slope = m = y x = y y = f x f(x ) = f t f(t ) = Average Rate of Change x x x x t t Applying the concept of limit to analyze and interpret the properties of functions (e.g. continuity, asymptotes). Applying the concept of a limit of a constant function: The limit of a constant function is the constant. We are able to see this fact by sketching a graph of f x = c lim c = c, Where c is any real number Vertical Asymptotes: A line y = b is a horizontal asymptote of the graph of the function y = f (x) from the left. lim f x = b Horizontal Asymptotes: A line y = b is a horizontal asymptote of the graph of the function y = f (x) from the right. f x = b lim Oblique Asymptotes: If the degree of the numerator of a rational function is 1 greater than the degree of the denominator, the graph has an oblique or slant line asymptote. To find the equation for the asymptote we divide the numerator by the denominator to express f as a linear function. After the division, if there exists a remainder, it goes to zero as x ± Applying the concept of rate of change to interpret statements from science, technology, economics, and other disciplines.
The concept of rate of change is applicable to many other disciplines: Physics: The displacement of the object over the time interval from t to t + t is s = f t + t f (t), Where s, is the displacement. The average velocity of the object over that time interval is v " = displacement travel time = s f t + t f (t) = t t Instantaneous velocity is the derivative of position with respect to time at a specific instant. If a body s position at time t is s= f (t), then the body s velocity at time t is: v t = ds dt = lim f t + t f(t) t This formula differs from the one above it since it is used to find the body s position at an exact instant and not over a designated period of time. Speed is the absolute value of velocity. Speed is a scalar, while velocity is a vector. Speed= v (t) = " Acceleration is the derivative of velocity with respect to time. If a body s position at time t is s = f (t), then the body s acceleration at time t is a t = dv dt = d s dt Business/Economics: In manufacturing operation, the cost of producing c (x) is a function of x, the number of units produced. The marginal cost of production is the rate of change of cost with respect to level of production, " ". Economists often, represent a total cost function by a cubic polynomial; c x = αx + βx + γx + δ In this function, δ represents the fixed cost, while the other terms represent variable costs, which include labor, materials, etc. It is important to note that these variables change from one problem to the next. Fixed costs are independent of the number of units produced, where variable costs depend on the quantity produced. Sensitivity of data has many business applications. A change in the data, whether dramatic or not can cause an impact. When a small change in x produces a large change in the value of the function, f x, we say the function is sensitive to changes in x. The derivative f (x) is a measure of this sensitivity. "
Examples: 1. Evaluate the following limit: lim " 2. Find the derivative of the following function: f x = 2x 16x + 35 3. Approximate the area under the curve f x = x + 2, 2 x 1 with Riemann Sums, using 6 sub intervals and right endpoints. 4. Evaluate the integral: (3x ) dx 5. Find the slope of the tangent line of the function f x = x 3 at (2,1) 6. Find the horizontal asymptote, x, of f x = 7. Determine the following for s = 6t t, 0 t 6 a. The body s displacement and average velocity for the given time interval b. The body s speed and acceleration at the endpoints of the interval c. When during the interval does the body change direction? 8. At t seconds after liftoff, the height of a rocket is 3t ft. How fast is the rocket climbing 10 seconds after liftoff? Resources: Thomas, B. G., Weir, D. M., & Hass, J. (2010). Thomas Calculus (12th ed.). Boston, MA: Pearson Education, Inc. http://tutorial.math.lamar.edu/classes/calci/defnofderivative.aspx http://clas.sa.ucsb.edu/staff/lee/secant,%20tangent,%20and%20derivatives.htm http://www.intuitive-calculus.com/fundamental-theorem-of-calculus.html http://www.lakelandschools.org/webpages/dcox/files/riemann%20sum%20practice.pdf http://www.tutorvista.com/content/math/discontinuity/