Calculus: The Basics. By: Rosed Serrano and Dominique McKnight. Table of Contents. About the Authors page 2 Chapter 1: Limits and Continuity page 3 Chapter 2: Derivatives page 8 Chapter 3: Anti-derivatives page 13 Chapter 4: Applications page 19. Rosed Serrano (left)
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Calculus: The Basics By: Rosed Serrano and Dominique McKnight
Table of Contents About the Authors page 2 Chapter 1: Limits and Continuity page 3 Chapter 2: Derivatives page 8 Chapter 3: Anti-derivatives page 13 Chapter 4: Applications page 19
Rosed Serrano (left) Rosed is a senior at the High School for Environmental Studies. Next year she will be attending Princeton University. While she is undecided on what she would like to study she is thinking about majoring in Economics. She is a leader and enjoys helping others. She enjoyed the challenges offered in AP Calculus and would like to continue taking higher level math courses. Dominique McKnight (right) Dominique is a senior at the High School for Environmental Studies. Next year she will be attending the University of Pennsylvania where she would major in Chemistry. She enjoys spending time with her friends. Her academic goals are to successfully attain her BS degree and work in the pharmaceutical industry. She enjoyed her time in AP Calculus as she understood the value of the class for college and her career. About the Authors
A limit is an approximate f(x) value for a given x value. A limit exists when there is an approximate f(x) value for an x value. For a one-sided limit to exits, f(x) has to approach a value as x approaches a certain value from either the left or the right. –OR- For a two-sided limit to exist the f(x) value approached when x approaches a value from the right and from the left must be equivalent. What does it mean for a limit to exist?
A limit fails to exist when the f(x) value approached when x approaches a value is equal to ∞ or -∞. A limit also fails to exist when the value of f(x) as x approaches a value from the right does not equal the value of f(x) when x approaches a value from the left. This can occur when the function is broken When do limits fail?
To solve using a calculator, first graph the equation. Then inspect the graph for any undefined x values. If the graph is smooth, you can find the limit by tracing f(x) as x approaches a value from the left and the right. You can also use the table of the graph and find the x values immediately less than and immediately greater than the x values you are approaching, once those values have been located from the average of their corresponding f(x) values. Solve using the calculator
Rule 1: f(x) = f(c) if f(x) is a polynomial function. This means that if you are solving for the limit of a polynomial function at x = c, you can just plug x = c into the function to find the limit Rule 2: The limit of a constant is the constant. Rule 3: The limit of a sum or difference of functions is equal to the sum or difference of the individual limits of each function. Rule 4: The limit of a product of functions is equal to the product of the individual limits of each function. Rule 5: The limit of a quotient of functions is equal to the quotient of the individual limits of each function, as long as you don't end up dividing by zero. In this case you will need to factor and reduce the equation before plugging in. Rule 6: To find the limit of a function that has been raised to a power, first find the limit of the function, and then raise the limit to the power. Algebraically solving limits
Continuity suggests that there are no holes or breaks in the function; for every x value there is a value for f(x). A function is not continuous if it does not meet those criteria. A gap in the function (usually indicated by a hole in the graph or in the domain of the function) means it is discontinuous, but it does not impact the limit because f(x) can approach a value it does not equal. That is to say f(x)≠2 but . One-sided limits also do not affect the continuity of a function as there is only one value needed, so a broken graph or an inconsistent domain does not matter. However for two-sided limits, the limit will never exist if the graph is fragmented, in other words if the limit from the right does not equal the limit from the left the graph is discontinuous and the (two-sided) limit does not exist. How limits impact the continuity of a function
The graphs of the first derivative and second derivative is very telling of the original function. The x-intercepts and the maximum or minimum of the graphs tells you about the way the curve in the original function looks. Remember these rules when graphing derivatives: 1. If the first derivative f'is positive (+) , then the function f is increasing. 2. If the first derivative f' is negative (-) , then the function f is decreasing . 3. If the second derivative f'' is positive (+) , then the function f is concave up . 4. If the second derivative f'' is negative (-) , then the function f is concave down. 5. The point x=a determines a relative maximum for function f if f is continuous at x=a , and the first derivative f' is positive (+) for x<a and negative (-) for x>a . The point x=a determines an absolute maximum for function f if it corresponds to the largest y-value in the range of f . 6. The point x=a determines a relative minimum for function f if f is continuous at x=a , and the first derivative f' is negative (-) for x<a and positive (+) for x>a . The point x=a determines an absolute minimum for function f if it corresponds to the smallest y-value in the range of f. 7. The point x=a determines an inflection point for function f if f is continuous at x=a, and the second derivative f'' is negative (-) for x<a and positive (+) for x>a , or if f'' is positive (+) for x<a and negative (-) for x>a. 8. THE SECOND DERIVATIVE TEST FOR EXTREMA (This can be used in place of statements 5. and 6.) : Assume that y=f(x) is a twice-differentiable function with f'(c)=0 a.) If f''(c)<0 then f has a relative maximum value at x=c. b.) If f''(c)>0 then f has a relative minimum value at x=c.. Graphing derivatives
To solve for derivatives algebraically, you must make sure you follow the rules listed below. Rules Common Functions Function Derivative Constant c0 x1 Power2x Example: Square Root √x ½x-½ f(x)=2 Exponential f’(x)=6(constant and power rule) (lna) Logarithmsln(x) 1/x loga(x) 1/(x ln(a)) Trigonometry (x is in radians) Rules Function Derivative FunctionDerivativeConstantcfcf’ sin(x) cos(x) Power Rule xnnxn-1 cos(x) -sin(x) Sum Rule f + g f’ + g’ tan(x) sec2(x) Difference Rule f - g f’ - g’ sin-1(x) 1/√(1-x2) Product Rulefg f g’ + f’ g tan-1(x) 1/(1+x2) Quotient Rule f/g (f’ g - g’ f )/g2 Reciprocal Rule 1/f -f’/f2 Chain Rule f(g(x)) f’(g(x))g’(x) Algebraically solving derivatives
Definition. Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit provided this limit exists. If this limit exists for each x in an open interval I, then we say that f is differentiable on I. Definition of derivatives
The derivative is the instantaneous rate of change or the slope of a function. An integral is the area under some curve between the intervals of a to b. An integral is like the reverse of the derivative, for instance, derivatives bring functions down a power while integrals bring them up. Relationship between the derivative and ant derivatives/ integrals
A function F is an anti-derivative or an indefinite integral of the function f if the derivative F' = f. We use the notation to indicate that F is an indefinite integral of f. Using this notation, we have If and only if Indefinite integrals
Properties of definite integrals: 1. 2. if c is a constant. 3. 4. 5.If 0 < f(x) < g(x) for all x in [a, b], then 6. If a < b then it is convenient to define. Definite integrals
Riemann sum is a summation of a large number of small partitions of a region. It may be used to define the integration operation. A Riemann Sum of f over [a, b] is the sum Right Riemann sum Right Riemann sum uses the right hand of the interval to find the area under the curve. For a curve that is strictly increasing the right Riemann sum will give you the overestimate of the area. For a function that is strictly decreasing the right Riemann sum gives you an underestimate of the area. Approximation methods
Left Riemann sum The left Riemann sum uses the left hand of the curve to find the are under the curve. For a curve that is strictly increasing the left Riemann sum is an underestimate. For a function that is strictly decreasing, the left Riemann sum gives you an overestimate. Midpoint sum The midpoint sum uses the middle of the curve to find the area under the curve
The trapezoidal sum uses the sum of the area of trapezoids that fit under the curve. This method is the average of the right Riemann sum and the left Riemann sum. This method can give you an over estimate or underestimate depending on the function. It gives you the value closest to the actual area under the Trapezoidal approximation Trapezoidal sum
1. An object's velocity, v, in meters per second is described by the following function of time, t, in seconds for a substantial length of time … v = 4t (4 − t) + 8 Assuming the object is located at the origin (s = 0 m) when t = 0 s determines … The object's position, s, as a function of time The object's acceleration, a, as a function of time The object's maximum velocity If and when the object stops If and when the object returns to the origin (s = 0 m) CALCULUS AND PHYSICS
The tide comes in every hour at Avalon Beach, from 0<t<5. Billy and his mom want to measure how far back they need to sit to avoid the water. At t=0 hours the water is already 4 inches on the shore. How far in inches will Billy and his mom have to sit from the shore in order to avoid the water if the rate at which the water approaches the shore is given by f(t) =2-3 in inches per hour. Step 1: Set up the integral. Don’t forget your initial condition! Step 2: Use your integration rules to solve the integral. This problem calls for the power and constant rule. Step 3: Simplify and solve 4+inches CALCULUS IN THE REAL WORLD
SparkNotes Editors. (n.d.). SparkNote on Functions, Limits, Continuity. Retrieved May 17, 2013, from http://www.sparknotes.com/math/calcab/functionslimitscontinuity/ Citation