Math 105 Exit Exam Review

Transcription

1 Math 105 Exit Exam Review The review below refers to sections in the main textbook Modeling the Dynamics of Life, 2 nd edition by Frederick F. Adler. (This will also be the textbook for the subsequent courses Math 204 and Math 205. Check with the department for the current edition being used.) The section on trigonometry is based on the set of Trig Notes (Parts 1 5). Section 1.2 Know the difference between parameter (fixed for an experiment but can vary from experiment to experiment) and variable (changes during course of experiment) Know how to graph data given in a table of values or as an equation of a function Know the difference between independent (input) and dependent (output) variable Know how to graph a function from a description and to describe a function from its graph Review Problems: 2, 37, 40, 41, 43, 48, 49 Section 1.3 Know how to convert from units in a given dimension to other units (miles to km etc.) by multiplying the given unit repeatedly with factors of 1 that come from conversion equations Know how to translate between dimensions using fundamental relations (for example mass and density) Know how to check the validity of an equation by checking the units Review Problems: 5, 13, 19-22, 31 Section 1.4 Know the slope-intercept and point-slope formulas for straight lines Know how to compute the slope of a straight line from two points Know that linear functions are equations of straight lines Know how to graph a linear function Know that two straight lines can have no, one or infinitely many intercepts and how they occur Know how the slope relates to increase or decrease of a linear function Know how to interpolate or extrapolate from a linear function Know that linear functions have constant slope = change in output/change in input Know that linear functions of the form () = are proportional, that is output/input is constant and the straight line passes through the origin Know how to determine whether a function is linear Know how to solve equations of linear functions Review Problems: 1, 7, 9, 13, 15, 43, 45, 53; Examples

2 Section 1.5 Know how to derive an updating function from a table of data (look at differences between output values or ratios between output values) Know that the updating function gives one time step, and that the composition of the updating functions k times gives the k-step updating function Know that the inverse function of the updating function computes a step backwards in time Know how to graph a solution in the time series graph (independent variable is time t, dependent variable is measurement at time t) Know the formula for the solution when the updating function is linear (not explicitly derived in the text): o If = then () = (0) o If = + then = 0 + if 1 = 0 + if = 1 Know how to read off the updating function from the DDS Review Problems: 1, 5, 9, 15, 19, 23, 27, 35, 37, 57, 58 Section 1.6 There will be no questions on cobwebbing. However, you will need to know how to determine equilibrium values. Know the equilibrium condition m*=f(m*) and how to solve for the equilibrium Know that linear functions can have either no, one, or infinitely many equilibrium values; biological apps either have one or none. Know that an equilibrium can be found graphically as the intersection of the updating function and the line x = y. Review Problems: 7, 19, 24 Section 1.7 Know the basic laws of exponents and logarithms and how to use them to solve an exponential or logarithmic equation Know how to convert from any base to base : " () = Know how to solve for the time at which the population has a given value, specifically doubling time and half-life Know how to determine the per capita production from half-life Review Problems: 35, 37, 43, 45, 49, 50

3 Trigonometry Notes (Parts 1-5) Know positive (counter-clockwise) and negative (clock-wise) orientation of angles Know how the different angle measures (degree, radian and revolution) are defined Know how to convert one angle measure into the other Know the Pythagorean Theorem and how to use it to compute the third side of a right triangle from the other two sides Know how the six trigonometric ratios are defined and be able to compute them from a given right triangle. Know the definition of the trig ratios in terms of the values of sin and cosine Know how to derive the values for sine and cosine of 30, 45, and 60 degrees (special triangles) Know the equation of the unit circle + = 1 and how to use it to determine whether a point is on the unit circle or not Know that the coordinates of a point on the unit circle are, = (cos, sin ) Know that a point on the unit circle, associated with angle has coordinates (cos, sin ) for all angles, extending from the right triangle for acute angles Know that he trig functions of all angles can be computed from those of the reference angle, with the sign determined by the quadrant. Know which trig functions are positive in each quadrant: All Students Take Calculus Be able to find the exact value for sine and cosine for a given special angle and their associated angles Compute the trig functions for an angle from its associated terminal point. Understand how the graphs of sine and cosine are derived from the unit circle definition Know that the graphs of sine and cosine repeat every 2 Know that the values of sine and cosine graphs are between -1 and 1. Be able to graph the basic sine and cosine functions. Review Problems: All problems listed in Trig Notes Part 5. Section 1.8 Know the formula of the general cosine curve = + cos the parameters ( ) and the meaning of A = average (vertical shift); B =amplitude (vertical scaling); T= period (horizontal scaling); =phase shift (horizontal shift) Know how to compute these parameters from given data to find a cosine curve that models the data Know how to read off these parameters from a given graph Know how to graph a given general cosine function from the formula. Maxima occur at beginning and end of period, the minimum occurs halfway into a period, and the average is crossed at 1/4 and 3/4 of a period. Review Problems: 33, 34, 37, 39, 41, 43

4 Section 1.9 You do not need to know the model of gas exchange in the lung. Know how to compute a weighted average (Definition 1.15 and Examples ) Know how to set up a mixing problem (Examples 1.9.5, 1.9.6) Review Problems: 1-4 Section 6.1 Read Examples and for motivation of stochastic models. Ignore discussion of histograms. Review Problems: 13, 15 Section 6.2 Know the diffusion model of molecules in and out of cells Know the meaning of diploid, allele, heterozygous, and homozygous. Know how to compute the genotype of the offspring from that of the parents using a table. Know the inheritance models of self-pollination and Mendelian genetics of dominant genes for plants that cannot self-pollinate Know how to compute the genotypes of second and higher generations taking into account the proportion of each genotype and the probability that it produces offspring of a given genotype Review Problems: 1, 3, 11, 12, 15, 16, 31 Section 6.3 Know definitions of experiment, sample space, simple event, event, and what it means for an event to occur. Also know when two events are mutually exclusive. Know how to compute the union and intersection of two events and also the complement of an event. Know the basic rules for probabilities and how to apply them. o Given probabilities for simple events, compute probabilities for events o o Given the probability for an event, compute the probability of its complement. Given probabilities for events A and B, compute probabilities of the intersection or union. Review Problems: 1, 3, 5, 9, 17, 24

5 Section 6.4 Know how conditional probability is defined: = ( ) () Know how to compute conditional probability by "shrinking the sample space" Know the multiplicative formula for the intersection of events (from definition of conditional probability): ( )=P(A B)P(B)=P(B A)P(A) If A and B are mutually disjoint (have empty intersection), then P(A B)=0 If B is contained in A, then P(A B)=1 Definition of partition (mutually exclusive and collectively exhaustive) Law of total probability in intersection form and conditional probability form: For a partition,,, and any event A: = = ( ) Bayes' Theorem: For any A and B where P(A) > 0, = () () Note that P(A) may be computed using the law of total probability. Review Problems: 2, 5, 29, 31, 33, 37, Independence Know the definition of independence Know the multiplication rule for independent events Know how to check for independence Review Problems: 1, 8, 33, 34