Derivative Rules and Rates of Change in Basic Differentiation

Dec 25, 2024

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Learn about the Constant, Power, Constant Multiple, Sum and Difference rules in basic differentiation. Discover how to find derivatives of trigonometric and exponential functions efficiently.

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Derivative Rules and Rates of Change in Basic Differentiation

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Section 3.2 Basic Differentiation Rules and Rates of Change
The Constant Rule The derivative of a constant function is 0. In other words, if c is a real number, then
The Power Rule: If n is a rational number, then the function is differentiable and (be careful at x=0) The Constant Multiple Rule: (if f is a differentiable function.)
The Sum and Difference Rules If f and g are differentiable, then their sum and difference are also differentiable. Furthermore,
Trig and Exponential Derivatives

Section 3.2

Section 3.2. Solving Systems Algebraically. Solving Systems Algebraically. ALGEBRA 2 LESSON 3-2. (For help, go to Lesson 1-1 and 1-3.). Find the additive inverse of each term. 1. 4 2. – x 3. 5 x 4. 8 y Let x = 2 y – 1. Substitute this expression for x in each equation.

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Section 3.2

Section 3.2. AP Statistics October 1 st , 2010 Mr. Calise. Correlation. Is there a “correlation” between a baseball team’s “earned run average” and the number of wins? Is the association strong or weak? Is the association positively associated or negatively associated?.

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Section 3.2

Section 3.2. Objective : SWBAT discern between the different types of specific crimes against people. Do Now Question . What are the types of crimes against people?. Objective : SWBAT discern between the different types of specific crimes against people Crimes Against People .

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Section 3.2

Section 3.2. Linear Programming Problems. Linear Programming Problem. A linear programming problem consists of a linear objective function to be maximized or minimized, subject to certain constraints in the form of linear equalities or inequalities.

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Section 3.2

Section 3.2. How Can We Describe the Spread of Quantitative Data?. Orchard hill: $125k, $132k, $144k, $156k, $452k River district: $178k, $189k, $204k, $213k, $227k We prefer to use the mean as a measure of center since it considers every number …

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Section 3.2

Section 3.2. How Can We Describe the Spread of Quantitative Data?. Range. One way to measure the spread is to calculate the range. The range is the difference between the largest and smallest values in the data set ; Range = max  min The range is strongly affected by outliers.

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Section 3.2

Section 3.2. Polynomial Division; The Remainder and Factor Theorems. Dividing Polynomials. Methods Long Division Synthetic Division. Quick Method of dividing polynomials Used when the divisor is of the form x – c Last column is always remainder. Synthetic Division. Remainder Theorem.

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Section 3.2

Section 3.2. Measures o f Variation. A little info…. Sometimes the measures of central tendency are not enough….. 2 different sets of data could have the same mean, but the data values vary drastically. So we need a different way to discuss the data.

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Section 3.2

Section 3.2. Applications of Linear Equations. GROWTH AND DECAY. In previous mathematics courses you have studied growth and decay problems. Some examples are population growth, radioactive decay, and carbon dating. The differential equation for growth and decay is

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Section 3.2

Section 3.2. Conditional Probability and the Multiplication Rule. Larson/Farber 4th ed. Section 3.2 Objectives. Determine conditional probabilities Distinguish between independent and dependent events Use the Multiplication Rule to find the probability of two events occurring in sequence

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Section 3.2

Math in Our World. Section 3.2. Truth Tables. Learning Objectives. Construct truth tables for negation, disjunction, and conjunction. Construct truth tables for the conditional and biconditional. Construct truth tables for compound statements. Identify the hierarchy of logical connectives.

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Section 3.2

Section 3.2. Norm of a Vector; Vector Arithmetic. PROPERTIES OF VECTOR ARITHMETIC. Theorem 3.2.1 : If u , v , and w are vectors in 2- or 3-space and k and l are scalars, then the following relationships hold. (a) u + v = v + u (b) ( u + v ) + w = u + ( v + w )

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Section 3.2

Section 3.2. Sequences and Summations. Sequence. Function from a subset of Z (usually the set beginning with 1 or 0) to a set S a n denotes the image of n (n  Z) a n is called a term of the sequence The notation {a n } is used to describe the sequence. Example.

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Section 3.2

Section 3.2. The Derivative as a Function. THE DERIVATIVE AS A FUNCTION. Given any number x for which the limit exists, we define a new function f ′( x ) , called the derivative of f , by. NOTE : The function f ′ can be graphed and studied just like any other function.

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Section 3.2

Section 3.2. Two players: The Divider-Chooser Method. The best known of all continuous fair-division methods Can be used anytime there is a continuous fair-division problem. Division. Player 1: (The Divider) – divides the cake into two pieces that he feels are fair shares

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Section 3.2

Section 3.2. The Mean Value Theorem. ROLLE’S THEOREM. Theorem : Let f be a function that satisfies the following three hypotheses: 1. f is continuous on the closed interval [ a , b ]. 2. f is differentiable on the open interval ( a , b ). 3. f ( a ) = f ( b ).

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Section 3.2

Section 3.2. Conditional Probability. Independent or Dependent Events. Two events A and B are independent if event B is not affected by the occurrence of event A. A = 1st child is a boy B = 2nd child is a boy. A = Being female B = Having brown hair.

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Section 3.2

Section 3.2 . Traits and Inheritance. VOCABULARY. Gene: One set of instructions for an inherited trait. Allele: One of the alternative forms of a gene that governs a characteristic, such as hair color. Phenotype: An organism’s appearance or other detectable characteristic.

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Section 3.2

Section 3.2. Section 3.3. Section 3.4. Section 3.5. Section 3.5 continued.

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Section 3.2

Section 3.2 . Finding Part. The Components of a Percent Problem. 1. BASE : the whole or total, starting point, or that to which something is compared 2. RATE : the percent (%) 3. PART : The result of multiplying the base and rate. PART of the whole. The Percent Formula.

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Section 3.2

Section 3.2. Crimes Against People. Some crimes can be specifically categorized as crimes against people. Murder. Murder is the unlawful killing of another human being with malice aforethought. Manslaughter.

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Section 3.2

Section 3.2. Graphing Linear Equations Using Intercepts. Objective 1. Use a graph to identify intercepts. Linear Functions. straight lines when graphed, as long as A and B are not both zero . Such an equation is called the standard form of the equation of the line .

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