LIMITS, CONTINUITY and DERIVATIVES
LIMITS
Unit Resources:
LIMITS
- Use the concept of a limit to find tangent slope.
- Use algebraic skills to determine the value of a limit.
- Solve problems using limits.
- Find limits to infinity.
- Use three conditions to determine if a function is continuous
- Use a mathematical process to determine a new function which can be used to find the slope of a given function.
Unit Resources:
- Limits and Derivatives Daily Plan
- Limits & Continuity Notes Pkg
- Derivatives Notes Pkg
- Calculus - A First Course, James Stewart. McGraw-Hill Ryerson, 1989. Chapter 1 Chapter 2
Assignments, Past Quizzes and Exams:
- Limits and Derivatives Exam:
- 2024: Exam and Answer Key
- 2023: Exam A and Answer Keys
- Limits Checks for Understanding:
- Limits 1: original and answer key
- Limits 2: original and answer key
- Limits 3: original and answer key.
- Derivatives Check for Understanding:
- Derivatives 1: Original and the Answer Key
- Derivatives 2: Original and the Answer Key
- Derivatives 3 (old quiz): Original and the Answer Key
- Quizzes
- Limits Quiz 2024: Quiz and Answer Key
- Derivatives Quiz 2024: Quiz and Answer Key
- Limits Quiz 2023: Quiz A and Answer Keys
- Derivatives Quiz 2023: Quiz A and Answer Keys
Base Skills for Calculus:
- Algebra Review: factoring, solving equations, exponents. Original and Answer Key
- Rational Review: - simplifying and solving. Original and Answer Key
- Factoring and Radical Skills:
- Factoring and Radicals HW - more factoring review; rationalize numerators in radical expressions.
- Original and Answer Key
- Textbook Questions: Calculus - A First Course, James Stewart. McGraw-Hill Ryerson, 1989. [page 3 and 4]
- page 3: factoring skills (1aceg, 2aceg, 3acef, 4acef)
- page 4: rationalizing skills (1abc)
LIMITS:
Use algebraic skills to determine the value of a limit (the function value we approach from both sides of 'x' but never get to; find the y-value for the open circle).
Use algebraic skills to determine the value of a limit (the function value we approach from both sides of 'x' but never get to; find the y-value for the open circle).
- Limits Lesson
- Examples
- Videos: Algebraically Determine Limits (zero divide by zero)
- Textbook Questions: Calculus - A First Course, James Stewart. McGraw-Hill Ryerson, 1989. page 19
- Limits Check for Understanding 1
DISCONTINUITY:
- Interpret the graphs of functions and identify points of discontinuity.
- Use limits to evaluate a function as you approach from one side - either the left side or the right side of an x-value.
- There are three conditions for being continuous: the limit as you approach the x-value exists, the function exists for the x value (the point exists), and the limit equals the function value at x (limit equals the point).
- For a Limit to Exist: We evaluate the one sided limits of the function (left and right side functions based on the domain). Limits exist at 'x' if we approach the same number (same y-value) from both sides of 'x'. [Do the instructions from the left side and instructions from the right side send you to the same location?]
- Points exist where we have the 'solid dot' for the function.
- If the first two conditions both exist, we determine if the limit equals the function and justify continuity.
- One Sided Limits and Discontinuity Lesson
- Examples
- Videos
- Textbook Questions: Calculus - A First Course, James Stewart. McGraw-Hill Ryerson, 1989. [page 27: 1,2,4,5,6,7]
- Examples
Use the concept of a limit to determine the tangent slope to a function.
Explore two limit formulas that can be used to calculate slope at a point on a function.
- Slope is rise over run, or y2-y1 over x2-x1. We substitute the known point x and y values in for x1 and y1. We use the general values of x and y from the function to complete the slope formula. We change the slope formula to a limit formula, looking to find the slope as we move the second point closer and closer to the first point (our given point).
- Warmup: Skills Review
- Limits to Find Slopes Lesson
Explore two limit formulas that can be used to calculate slope at a point on a function.
- Slope is rise over run. Tangent slopes are found using the normal slope formula, modified to allow for:
- making two points overlap at the point of tangency
- making the horizontal distance between the two points equal zero (first principles).
- Tangent Slopes Lesson
- Examples
- Textbook Questions: Calculus - A First Course, James Stewart. McGraw-Hill Ryerson, 1989. [Page 35 - 7, 8]
- Examples
- Limits Check for Understanding 2
- Limits Check for Understanding 3
INFINITY:
To determine limits at infinity we need to work with two concepts
Limits to Infinity of Rational Functions (where the degree of the denominator is equal to or larger than the degree of the numerator; we can create a pattern similar to a convergent sequence combined with a constant function). To evaluate rational functions at infinity, we use some reverse thinking. For limits to a specific number, we used math to take out fractions in numerators and denominators. The reverse is true for rational functions: To evaluate a rational function at infinity we need to put 1/x or 1 over x to a higher degree into the rational function allowing us to use the zero property and to use the limit of a constant property.
To determine limits at infinity we need to work with two concepts
- Are the values of a sequence convergent? Convergent sequences have a common ratio |r|<1; where consecutive terms get smaller.
- What is the value of the function f(x)=1/x as the value of x approaches infinity. As a number divides by a larger and larger number, the quotient becomes smaller and smaller; eventually becoming so small we say we have reached a quotient of zero.
- The function is a constant. Example f(x)=3. The function is always approaching the same value from left and right sides (in our example, both sides approach 3) so the limit of a constant function at infinity is the constant.
- If the function is a rational. Example f(x)=1/x. The limit of a function where we divide a number by x or divide by x to a higher degree is zero.
Limits to Infinity of Rational Functions (where the degree of the denominator is equal to or larger than the degree of the numerator; we can create a pattern similar to a convergent sequence combined with a constant function). To evaluate rational functions at infinity, we use some reverse thinking. For limits to a specific number, we used math to take out fractions in numerators and denominators. The reverse is true for rational functions: To evaluate a rational function at infinity we need to put 1/x or 1 over x to a higher degree into the rational function allowing us to use the zero property and to use the limit of a constant property.
- Limits to Infinity for Rational Functions Lesson
- Examples
- Video 1: Graphs and properties
- Video 2: Rational Functions at Infinity
- Textbook Questions: Calculus - A First Course, James Stewart. McGraw-Hill Ryerson, 1989. [Page 50]
- Examples
DERIVATIVES:
Derivatives - the process of finding a slope function.
If we have an equation to find slope, we can substitute the x value for any point on the function and determine the tangent slope at that given value on the function. Knowing the slope and the point of the function, we can find the equation of the tangent line at any point on the function.
We found the slopes of 11 quadratic functions and 11 cubic functions using limits and h approaches zero (first principles). From these examples we began to see patterns for slopes:
- The slopes of quadratic or cubic functions don't change if they only differ by a vertical translations. If we have a stretch on the function, the slope equation has the same stretch factor applied.
- We also noticed if the original function is a sum (or difference) of terms, then the slope function is the sum (or difference) of each individual term.
Basic Derivative Rules
Constant Rule: Think about the slope of y=5 or y=12 or y=-2; the slope for any horizontal line is zero. If we need to find the derivative for a constant term: dy/dx(constant) = zero.
Power Rule: Looking at the pattern on polynomial functions using limits (h approaches zero) we see:
Sum and Difference Rule:
The derivative of a polynomial function with more than one term can be found by applying the Power Rule or Constant Rule to each individual term.
We need to practice the rules and then be able to apply problem solving skills involving slopes. Example 5: We know we can find tangent slope to any point on the parabola using the derivative slope. We know we can find the slope of a line through a know point using delta 'y' over delta 'x'. Since the two slopes are for the same line, we make the derivative slope function equal the line slope function and solve for x. When solving for x, we need to use only x's in our equation for slope of the line; as a result we substitute y=2x-x^2 into this equation.
Constant Rule: Think about the slope of y=5 or y=12 or y=-2; the slope for any horizontal line is zero. If we need to find the derivative for a constant term: dy/dx(constant) = zero.
Power Rule: Looking at the pattern on polynomial functions using limits (h approaches zero) we see:
- The slope function is always one degree less than the given function.
- The Original Degree of the function gets multiplied by the vertical stretch.
Sum and Difference Rule:
The derivative of a polynomial function with more than one term can be found by applying the Power Rule or Constant Rule to each individual term.
We need to practice the rules and then be able to apply problem solving skills involving slopes. Example 5: We know we can find tangent slope to any point on the parabola using the derivative slope. We know we can find the slope of a line through a know point using delta 'y' over delta 'x'. Since the two slopes are for the same line, we make the derivative slope function equal the line slope function and solve for x. When solving for x, we need to use only x's in our equation for slope of the line; as a result we substitute y=2x-x^2 into this equation.
Product Rule:
The derivative of a product (two functions being multiplied) is NOT found by multiplying the individual derivatives. The pattern for product rule is: Multiply the derivative of the first function by the second function, multiply the first function by the derivative of the second function AND add up the two products.
Quotient Rule:
The pattern for the quotient rule can be found by changing the quotient of two functions to a product of two functions. This process requires way to much work with negative exponents. Because there is a similarity between multiplying and dividing, the quotient rule looks similar to the product rule - except the pattern is to subtract and we also have to divide by the square of the denominator (if a quotient is written as a product, the denominator function exponent would be written as negative one; applying the power rule, decrease by a degree, this exponent would become negative two, which is the same as squared function back in the denominator.)
Chain Rule:
If we have a composition of functions, we use the chain rule to simplify. For many of our early examples, this is a two step process:
Combination of Rules:
Some questions will require the use of the chain rule in conjunction with other rules, there will even be questions where we use the chain multiple times. We need to apply the necessary rules to the 'layer' of the questions.
The derivative of a product (two functions being multiplied) is NOT found by multiplying the individual derivatives. The pattern for product rule is: Multiply the derivative of the first function by the second function, multiply the first function by the derivative of the second function AND add up the two products.
Quotient Rule:
The pattern for the quotient rule can be found by changing the quotient of two functions to a product of two functions. This process requires way to much work with negative exponents. Because there is a similarity between multiplying and dividing, the quotient rule looks similar to the product rule - except the pattern is to subtract and we also have to divide by the square of the denominator (if a quotient is written as a product, the denominator function exponent would be written as negative one; applying the power rule, decrease by a degree, this exponent would become negative two, which is the same as squared function back in the denominator.)
Chain Rule:
If we have a composition of functions, we use the chain rule to simplify. For many of our early examples, this is a two step process:
- Apply the Power Rule to reduce the 'base function degree' by one, multiply by degree too.
- Take the derivative of the 'base function.
Combination of Rules:
Some questions will require the use of the chain rule in conjunction with other rules, there will even be questions where we use the chain multiple times. We need to apply the necessary rules to the 'layer' of the questions.
Implicit Differentiation:
Use the chain rule to find the derivative of any 'y' value in the relation. To find the slope in a relation, we generally need to know both the x coefficient and y coefficient for the tangent point.
Higher Order Derivatives:
In many ways higher derivatives are not a difficult concept - take the derivative of the derivative and repeat as necessary. The challenge is with the simplifying of the derivatives before moving on to the next step. We need to use fraction skills, factoring skills, simplifying skills for many of these problems.
Use the chain rule to find the derivative of any 'y' value in the relation. To find the slope in a relation, we generally need to know both the x coefficient and y coefficient for the tangent point.
Higher Order Derivatives:
In many ways higher derivatives are not a difficult concept - take the derivative of the derivative and repeat as necessary. The challenge is with the simplifying of the derivatives before moving on to the next step. We need to use fraction skills, factoring skills, simplifying skills for many of these problems.