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UTM / MAT134

2300

Transforming Graphs Question: How does the graph of relate to the graph of ? Consider, We see that the graph is SHIFTED to the left by three units. Question: How does the graph of relate to the graph of ? Consider, In general , the graph of is the graph of shifted: To the left if is positive To the right if is negative Vertical Shifts Question: How does relate to ? Consider the example, The graph is vertically shifted up one unit. Question: How does relate to ? Con

700

Recall signed area of on the interval We generalize this idea to the area between two curves. Example - Find the area between on Area = Area under - area under Example - Find the area between on Area Example - Find the area between on , Find points of intersection is above on on Area Example - Find the area between on Find the points of intersection Area Horizontal Slices General

1100

Concavity Define Let be differentiable. is concave up if is increasing concave down if is decreasing Concave up Concave down Example The curve is concave up The curve is concave down Discuss Determine where is concave up and concave down Differentiate f(x) Determine where is increasing/decreasing Thus, f(x) is concave up on and concave down on The Second Derivative Test for Concavity If the f(x) is concave up. If then f(x) is concave down. Example Find the interva

800

Recall is an anti-derivative of if: Note - We write for the family of anti-derivatives of Facts LInearity We can find anti-derivatives by using a table of derivatives backwards. Substitution Recall We obtain Example Example The U-Substitution Method To integrate (1) Let (2) Integrate (3) Undo the Substitution by Example Example In general, we might have to do some algebra before subs

UTM / BIO152

3600

Asymmetry of Sex Sperm - Inexpensive, many, small Egg - Large, expensive, few Females Usually invest more in their offspring than males do Typically produce relatively few offspring Fitness is limited primarily by the ability to gain resources necessary to produce/rear young Female Choice The behaviour of a female in choosing a mate that biases the mating success of the males toward the preferred type Preference depends on phenotype of males Female moves towards the male with preferred male Ta

6100

Natural Selection Darwin’s Process of evolution by natural selection Variation in traits Some trait differences are heritable More offspring are produced than can survive Some will produce more offspring than others Natural selection occurs when individuals with certain traits produce more offspring than do individuals without those traits These selected traits will increase in frequency in the population from one generation to the next, causing evolution Heritable variation leads to differenti

2100

Derivatives and Inverses Recall, the inverse of a function is a function such that: EXAMPLE Find the inverse of Solve for in terms of Switch and Define Check: Thus, is the inverse of . The notation is very common. QUESTION If then how are and related? (HINT: Use the chain rule) FACT DISCUSS If find Recall, KEY FACT EXAMPLE Find and DISCUSS Calculate using a calculator and at and compare with EXAMPLE Suppose

2200

Derivative as Function Recall from last lecture, The SLOPE of at is This function of is ''derived ' from and we call it the DERIVATIVE of Example Stretch the derivative of Example Calculate if Notation If we write: '"Leibnitz" "Newton" Example Find if Clear denominators Example Find if Conjugate the top One Sided DEfinition

Rate Of Change Consider the following You are driving at 120 km/hr. What does this mean? Discuss How far do you travel in 2hr? 1hr? 30 mins.? 1 min.? Two photos taken 1 sec. apart show that a car has moved 29m. How fast? Define The AVERAGE RATE OF CHANGE of between and is Discuss What is the average rate of change of from to ? ? Speed Rate of change of distance Instant Rate of Change Example An apple falls with displacement , how fast is it travelling at ?

1500

So far we have learned the following facts about anti-derivatives or indefinite integrals If then FTC Example - Today we will extend our set of tools for working with indefinite integrals. Further Trig Integrals Example - Find We obtain Example - Find It follows that Example - Find Therefore Inverse Trig Functions Recall, We obtain this by calculating How do we The term that

1400

Recall - notation Definition - Given a sequence: we define the sequence of PARTIAL SUMS The sum is an INFINITE SERIES We say CONVERGES if: converges Note Sequence Numbers Series Sum To get a sense of what series are all about consider the following examples However, Algebra of Series Notation - We write Fact - If and then Ex. If and find Fact - If diverges then diverges for any The Term Divergence Test Fact - If conver

Recall, the comparison test for integrals Suppose for all diverges diverges converges converges Example #1 Determine whether diverges The function resembles We observe Thus, This converges since and we may apply the test Fact (Comparison for Series) Suppose for all diverges diverges converges converges example #2 Show that converges example #3 Determine whether converges or diverges We have Thus, converges Recursive Sequences A sequence is recur

UTM / BIO153

14100

Learning How to Preserve Biodiversity To get a complete understanding of the diversity of life, biologists recognize and analyze biodiversity on 3 levels 1) Genetic diversity is the total genetic information contained within all individual of a species and is measured as the number and relative frequency of all ales present in a species 2) Species diversity based on the variety of species on earth measured by quantifying the number and relative frequency of species in a region taxonomic diversit

1700

The Idea Of Limits Limits are a tool used in calculus to learn about the value of a function at by studying the values near Example Complete the following tables of Values: It looks like is close to and so we expect Near the function is near Definition If is arbitrarily close to when is sufficiently close to then THE LIMIT OF AS APPROACHES IS and we write Mainly, " is close to when is close to " Example Calculate Consider If is close to then is c

UTM / PHY136

2500

Energy is conserved; can’t be created / destroyed Kinetic Energy describes motion => mv^2 Energy on earth originates from the sun Energy on earth is stored thermally and chemically Chemical energy is released by metabolism Energy is stored as potential energy in objects Conservation of Energy Energy cannot be created or destroyed, but can be transformed other energies of another form. Dissipation of “heat” – Energy could be lost by transforming into heat Work Work done is the dot product (scal

In Physics, Work is a measure of energy usage. To move an object distance the meters using force F newtons requires joules Example - How much work is required to lift 10kg up 2m on Earth? Gravity on Earth is Apply the work formula Example - How much work is required to lift 10kg up 2m on the moon? Gravity on the moon is Key Equations Force Work We use calculus to deal with situations where the mass/force is changing. Example - A rope weighs 1kg/m and is hanging 20m

Substitution Recall The Idea of Substitution is to integrate let (1) (2) (3) Undo the Substitution Often to apply substitution we need to introduce a constant factor. Example (Apply the substitution) (Introduce the constant 2) Example (Apply Substitution) (Introduce constant) (Undo the Substitution) We have to "spot" the right application of the chain rule to use Substitution. Example Sometimes Algebra

7710

An Overview of Animals Animals are a particular species rich and morphologically diverse lineage of multicellular organisms on the tree of life Animals are distinguished by several traits other than multicellularity, eating, and moving – The cells of animal’s lack cell walls but have an extensive extracellular matrix which includes proteins specialized for cell-cell adhesion and communication Animals are the only lineage on the tree of life with species that have muscle tissue and nervous tissue

UTM / ANT102

9311

Speech Sounds McLean Identified that the Brain was Made of Three Main Components Nonmammalian (neo-cortex) Found in higher mammals Most developed in humans Divided into left and right hemispheres Humans cannot live without neo-cortex (2/3 of the mass of brain) Very complex in structure Most of it grows after birth Paleomammalian (limbic system) Consists of several components Seat of social motivation (social brain) Involved in organized fighting, sex/reproduction, parental Decides whether higher

Partial Fractions When integrating ratios of polynomials we need to use some special techniques. One maneuver is to break up a fraction. Example - Solve for A and B We obtain Thus, Example - Discuss - Find A and B so that Using this decomposition we get Example - Express Clear fractions by multiplying and Thus, Discuss - Express Clear Fractions Thus, Example - Express Clear fractions We obtain