unit 7
1. consider the differential equation dy/dx = y^2/x, where x ≠ 0. a. on the axes provided, sketch a slope field for the given differential equation at the eight points indicated. b. find the particular solution y = f(x) to the differential equation with the initial condition f(1) = -1. c. write an equation for the tangent line to the curve y = f(x) through the point (1,-1). then use your tangent line equation to estimate the value of f(1.2) 2. the rate of change of the volume V(t) of water in a swimming pool is directly proportional to the cube root of the volume. if V = 27ft^3 when Dv/dt = 5, what is a differential equation that models this situation?
1. a. check it b. y = -1/(lnIxI + 1) c. y = -0.8; y + 1 = x - 1 2. dV/dt = 5/3(^3√V)
Euler's method is a numerical approach to approximating the particular solution of a differential equation that passes through a particular point. this method is useful when:
1. an approximate value of a function is desired 2. we know a given value of the function close to the desired value 3. we have the equation of the derivative 4. solving the differential equation analytically is difficult or impossible
1. find the values of b and k if L = 100 =, y(0) = 6 and y(1) = 15 2. find the limit of the population with y(0) = 20 and k = 0.5 and y(1) = 25 3. find the limit of the population with y(0) = 30 and y(1) = 40 and k = 0.5 4. find the particular solution to the differential equation dy/dt = 0.5y(1 - y/5000) if y(0) = 75
1. b = 15.67; k = 1.0169 2. L = 40.675 3. L = 82.28 4. y = 5000/(1 + 65.67e^(-0.5t))
1. the rate at which a quantity B of a certain substance decays is proportional to the amount of substance at a given time. which of the following could describe the relationship. 2. the rate of change of P with respect to t is proportional to (25 - t). write and solve the differential equation that models the situations
1. dB/dt = 2.4B 2. P = 25kt - kt^2/2 + c
write a differential equation that describes each relationship. if necessary, use k as the constant of proportionality. 1. the rate of change of F with respect to s is inversely proportional to x. 2. the rate of change of R with respect to t is proportional to the product of S and T. 3. the rate of change of R with respect to x is proportional to m and inversely proportional to the square of d. 4. kinetic energy, E, changes with respect to t at a rate that is proportional to the square of the velocity, v, and inversely proportional to the mass, m. 5. the rate of change of the volume, V(t), of a right circular cone with respect to time (in seconds) is increasing at a rate proportional to the product of the square of its radius, r, and its height, h. find the differential equation if the cone has a height of 8 inches, radius 3 inches, and the volume is changing by 2 cubic inches per second. 6. the maximum safe load, S, for a horizontal beam of fixed width changes depending on the length, l, of the beam. S decreases with (with respect to l) proportional to the product of its width, w, the square of its height, h, and inversely as its length, l. 7. mr. kelly is running down his street. his position is given by the function p(t), where t is measured in seconds since the start of his run. his acceleration is proportional to the square root of the time since the start of his run. his acceleration is 0.0277 km/sec/sec at 3 seconds. 8. the amount of time, T, that it takes fruit to ripen changes depending on the earth's latitude, L, where the equator is 0 degrees latitude. the rate of change of T, with respect to the latitude, is inversely proportional to the square of L. 9. mr. brust is outside singing one evening, and the neighborhood dogs chime in. the longer he sings, the more difficult it is for him to maintain a solid tone, his voice change can be modeled by the rate of change of frequency, F, with respect to time that is inversely proportional to D, the decibel level of his voice. if the frequency is changing by 3 vibrations per second when he is projecting at 80 decibels, what is a differential equation that describes this relationship? 10. the height of a rocket is given by the function h(t), where t is measured in seconds since the launch and h is measured in meters. the acceleration is proportional to the square root of the time since the start of the launch. at 21 seconds, the acceleration is 5 meters per second per second.
1. dF/ds = k/x 2. dR/dt = kST 3. dR/dx = km/d^2 4. dE/dt = k(v^2/m) 5. dV/dt = 0.0277r^2h 6. dS/dL = (kwh^2)/L 7. d^2P/dt^2 = 0.016√t 8. dT/dL = k/L^2 9. dF/dt = 240/D 10. d^2h/dt^2 = 1.091√t
set up a differential equation for each scenario. 1. the weight of an animal is increasing at a rate proportional to its weight 2. a bacteria population is shrinking at a rate proportional to its population size. 3. solve dy/dt = ky using separation of variables
1. dw/dt = kw 2. dP/dt = kP 3. y = ce^(kt)
write a differential equation that describes each relationship. if necessary, use k as the constant of proportionality. 1. the rate of change of y with respect to x is proportional to the product of t and the rate of change w with respect to x. 2. the force F of a spring on a trampoline can be related to the distance D it is stretched. the rate of change of the quantity of F with respect to the distance D is inversely proportional to the natural logarithm of the distance D. if the rate of change of F is 3 units per cm, then the spring has been stretched 0.2cm. what is a differential equation for this situation? 3. mr. brust is swimming in a straight line across a lake. his position from his starting point is given by p(t), where t is measured in minutes since the start of his swim. during the first 30 seconds, mr. burst's acceleration is proportional to the cube root of the since the start of his swim. write a differential equation that describes this relationship.
1. dy/dx = kt(dw/dx) 2. dF/dD = -4.828/ln(D) 3. P"(t) = d^2p/dt^2
find the general solution of each differential equation: 1. dy/dx = x^2/y 2. dy/dx = (sinx)y^2 3. dy/dx = xy 4. dy/dx = 2xy + 4x 5. dy/dx = ycosx 6. dy/dx = (y+5)(x+2) 7. dy/dx = e^(x-y) 8. dy/dx = 2x/e^(2y) 1. dy/dx = 3x^2/y 2. dy/dx = 8x^2y 3. dy/dx = e^xy^2 4. dy/dx = -2x(y-3)
1. y = +/-√(2/3x^3 + c) 2. y = 1/(cosx + c) 3. y = ce^(x^2/2) 4. y = ce^(x^2) - 2 5. y = ce^(sinx) 6. y = ce^(x^2/2 + 2x) - 5 7. y = ln(e^x + c) 8. y = ln(√(2x^2 + c)) 1. y = +/- √(2x^3 + c) 2. y = ce^(8/3x^3) 3. y = 1/(-e^x + c) 4. y = c/(e^(x^2)) + 3
find the general solution of the differential equation 1. dy/dx = 2x/y 2. dy/dx = x(y+4)
1. y = +/-√(2x^2 + c) 2. y = ce^(x^2/2 - 4)
find the particular solution y = f(t) for each differential equation. 1. dy/dt = 8y and y = -2 when t = 0 2. dy/dt = -4y and y = 10 when t = 0 3. dy/dt = 16y and y = 5 when t = 0 4. dy/dt = -7y and y = -4 when t = 0
1. y = -2e^(8t) 2. y = 10e^(-4t) 3. y = 5e^(16t) 4. y = -4e^(-7t)
1. if dy/dx = e^(2x) - 2x^2, find the particular solution of y if y(0) = 4 2. this problem is exactly the same. a curve has a slope of e^(2x) - 2x^2 at each point (x,y) on the curve. what is an equation for this curve if it passes through the point (0,4)? 3. if d^2y/dx^2 = 1/x^2 + (1-2x)^2, find the particular solution of y if y'(1) = 7/6 and y(1) = 0.
1. y = 1/2e^(2x) - 2/3x^3 + 3.5 2. *same as #1 just written differently* 3. y = -lnIxI + (1-2x)^4/48 + 2x - 97/48
for each differential equation, find the particular solution that passes through the given point. 1. dy/dx = 4x + 2; (-1,3) 2. dy/dx = 3/(2-x) + 6x^2; (1,1) 3. dy/dx = 8cos(4x); (π/8,-2) 4. dy/dx = 9e^(3x) - 1; (0,7) 5. d^y/dx^2 = 1/(2-x)^2 + 1 and y'(3) = 6 and y(1) = 4 6. d^2y/dx^2 = e^(2x) - x and y'(0) = 3/2 and y(0) = 3/4
1. y = 2x^2 + 2x + 3 2. y = -3lnI2-xI + 2x^3 - 1 3. y = 2sin(4x) - 4 4. y = 3e^(3x) - x + 4 5. y = -lnI2-xI + 1/2x^2 + 4x - 1/2 6. 1/4e^(2x) - 1/6x^3 + x + 1/2
LOOK AT THE GRAPHS FOR THESE PROBLEMS ON 7.7!!!! 1. find the solution to the differential equation dy/dx = (xy)^2 with initial condition y(1) = 1. 2. find the solution y = f(x) to the differential equation dy/dx = 2x/y with initial condition f(2) = 1 3. find the solution to the differential equation dy/dx = (y+2)e^x with the initial condition y(0) = -1
1. y = 3/(-x^3 + 4) 2. y = √(2x^2 - 7) 3. y = e^(e^(x) - 1) - 2
for each differential equation, find the solution that passes through the given initial condition 1. dy/dx = ycosx and y = 4 when x = 0 2. dy/dx = 6x^2/y if y(0) = -2 3. dy/dx = ysinx if y(π/2) = 2 4. dy/dx = 1/5(8-y) and y = 6 and when x = 0 5. dy/dx = (y+5)(y+2) when f(0) = 1 6. dy/dx = e^x/y if y(0) = -4
1. y = 4e^(sinx) 2. y = -√(4x^3 + 4) 3. y = 2e^(-cosx) 4. y = -2e^(1/5x) + 8 5. y = 6e^(-1/2x^2 + 2x) - 5 6. y = -√(2e^x + 14)
for each differential equation, find the particular solution that passes through the given point 1. dy/dx = 12/(3x-2) - 1/x^2; (1,-3) 2. dy/dx = 10sin(5x); (π/5,1) 3. dy/dx = 6e^(2x) - 4x; (0,-2)
1. y = 4lnI3x-2I + 1/x - 4 2. y = -2cos(5x) - 1 3. y = 3e^(2x) - 2x^2 - 5
identify the class of function, and find the general solution 1. dy/dx = -7 2. dy/dx = -7y 3. dy/dx = y - 7 4. dy/dx = 7x/y 5. dy/dx = -7x 6. dy/dx = x - 7 7. dy/dx = -7x/y 8. dy/dx = 2y(1 - y/10)
1. y = kx + c; linear 2. y = ce^(-7x); exponential 3. y = ce^(x) + 7; exponential vertical shift 4. y = -7x^2 + y^2 = c; hyperbola 5. y = -7x^2/2 + c; quadratic 6. y = x^2/2 - 7x + c; quadratic 7. y = 7x^2 + y^2 = c; ellipse 8. y = 10/1 + be^(-2t); logistic
11. of the following, which are solutions to the differential equation y" - 5y' + 4y = 0 I. y = 5cos(2x) II. y = 2e^x III. y = Ce^x, where C is a constant 12. consider the differential equation dy/dx = (y-4)^(3)sin(πx/2). there is a horizontal line with equation y = c that satisfies this differential equation. find the value of c.
11. II and III only 12. y = 4. a horizontal line exists only if the slope (derivative) is zero. this differential equation will equal zero when y = 4.
11. if s(t) is a baby's shoe size at time t, which of the following differential equations describes linear growth in the size of the baby's shoes 12. the rate at which a quantity B of a certain substances decays is proportional to the amount of the substance present at a given time. which of the following is a differential equation that could describe this relationship.
11. ds/dt = 2t 12. dB/dt = -0.32B
13. bacteria in a certain culture increase at a rate proportional to the number present. if the number of bacteria doubles in five hours, in how many hours will the number of bacteria triple? 14. a group of tiny organisms (measured in grams) is shrinking at a rate modeled by dM/dt = -0.7M, where the time t is measured in hours. the size of a second group of organisms decreases at a constant rate of 2 grams per hour according to the linear model dN/dt = -2. if at time t = 0, the first group has size M(0) = 4 and the second colony has size N(0) = 6, at what time will both groups of organisms be the same size?
13. 5ln3/ln2 14. 2.697
find the particular solution for each differential equation. 3. dy/dt = 6y and y = 5 when t = 0 4. dy/dx = -3y and y = 4 when x = 0 5. an animal weighs 3 pounds at birth and 4 pounds just three months later. the weight of the animal is increasing at a rate proportional to its weight. set up a differential equation for this scenario. how much will the animal weigh when it is 5 months old?
3. y = 5e^(6t) 4. y = 4e^(-3t) 5. 4.845 lbs
4. for what value of k, if any, will y = ke^(-4x) - 2sin(5x) be a solution to the differential equation y" + 25y = -82e^(-4x)
4. k = -2
5. P(t) = 2100/(1 + 29e^(-0.75t)) is a logistics equation that models growth of a population. identify the following: a. the value of k b. the carrying capacity c. the initial population d. determine when the population will reach 50% of its carrying capacity e. write the logistic differential equation for P(t) 6. find the logistic equation that satisfies the initial condition a. dy/dt = y(1 - y/36) initial condition (0,4) b. dy/dt = 2.8y(1 - y/10) initial condition (0,7)
5. a. 0.75 b. 2100 c. 70 d. 4.81 e. dy/dx = 0.75y(1 - P/2100) 6. a. y = 36/(1 + 8e^(-t)) b. y = 10/(1 + 0.429e^(-2.8t))
for each problem, use your understanding of exponential models and differential equations. 5. during a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. if 500 people are infected when the epidemic is first discovered, an 800 people are infected 5 days later, how many people are infected 10 days after the epidemic is first discovered? 6. a population y grows according to the equation dy/dt = ky, where k is a constant and t is measured in years. if the population doubles every 11 years, then what is the value of k? 7. a certain animal weighs 30 grams at birth. during the first 4 weeks after the animal's birth, its weight in grams is given by the function W that satisfies the differential equation dW/dt = 0.01W, where t is measured in days. what is an expression for W(t)? 8. a population y grows according to the equation dy/dt = ky, where k is a constant and t is measured in years. if the population doubles every 25 years then what is the value of k? 9. a population of fruit flies is increasing at an exponential rate. if on the 3rd day there were 400 fruit flies, and the 7th day there were 600 fruit flies, approximately how many flies were in the original population (day 0)? 10. a Petri dish contains 100 bacteria, and the number N of bacteria is increasing according to the equation dN/dt = kN, where k is a constant and t is measured in hours. at time t = 3, there are 181 bacteria. based on this information, what is the doubling time for the bacteria? 11. a radioactive substance has a rate of decay that can be modeled by the equation dy/dt = ky, where k is a constant and t is measured in years. if the half-life of the radioactive substance is 300 years, then what is the value of k? 12. a dose of 100 milligrams of a drug is administered to a patient. the amount of the drug, in milligrams, in a person's bloodstream at time t, in hours, is given by A(t). the rate at which the drug leaves the bloodstream can be modeled by the differential equation dA/dt = -.05A. write an expression for A(t).
5. about 1280 people 6. k = 0.063 7. w(t) = 30e^(0.01t) 8. k = 0.0277 9. 295 flies 10. t = 3.506 hours 11. k = -0.0023 12. A(t) = 100e^(-0.5t)
for each differential equation, find the particular solution that passes through the given point. 5. dy/dx = 18/(6x + 3) + 4/x^3; (-1/3, -15) 6. dy/dx = 2y and y. = -0.2 when x = 0
5. y = 3lnI6x + 3I - 2/x^2 + 3 6. y = -0.2e^(2x)
6. for what value of k, if any, will y = ke^(-3x) + 8sin(2x) be a solution to the differential equation y" + 4y = 26e^(-3x)? 7. for what value of k, if any, will y = kcos(3x) - sin(5x) be a solution to the differential equation y" + 25y = 8cos(3x)?
6. k = 5.2 7. k = 1/2
7. a population y grows according to the equation dy/dt = ky, where k is a constant and t is measured in years. if the population doubles every 12 years, then what is the value of k? 8. look at for the slope field (review) 9. for what value of k, if any, will y = cos(2x) + 3sin(4x) be a solution to the differential equation y" + 16y = -6cos(2x)?
7. k = 0.0578 8. when y = 0, dy/dt = 0. however, in the slope field, the slopes of the lien segments for y = 0 are nonzero. 9. k = -1/2
7. find the particular solution to dy/dx = e^(x-y) when f(0) = 2. sketch the graph of this particular solution on the slope field provided. 8. find the particular solution to dy/dx = xy^2 if y = 1 when x = 0. sketch the graph of this particular solution on the slope field provided
7. y = ln(e^x + e^2 - 1) 8. y = -2/x^2 + 1 look at the graphs on page 3 of 7.7
find the values of k of each equation that would be a solution to the given differential equation. 7. y = 3ke^(2x) + cos(4x) diff eq: y"/2 + 8y = 15e^(2x) 8. y = sin(-x) + 2cos(3x) diff eq: 2y" + 18y = 32sin(-x) 9. y = e^(-3x) = ke^(4x) diff eq: 3y' + y" = -14e^(4x) 10. y = e^(3x) + ke^(-2x) diff eq: y" - 2y' - 3y = 4e^(-2x)
8. k = 1/2 9. k = 2 10. k = -1/2 11. k = 4/5
14. the rate at which a baby koala bear gains weight is proportional to the difference between its adult weight and its current weight. at time t = 0, when the bear is first weighed, its weight is 2 pounds. if B(t) is the weight of the bear, in pounds, at time t days after it is first weighed, then dB/dt = 1/4(20 - B). find y = B(t), the particular solution to the differential equation.
B(t) = -18e^(-1/4t) + 20
the solution to dy/dx = ky is y =
Ce^(kt), where C represents the initial value of the model
look at all of 7.4 for why the reasoning behind using slope fields!!!!!
ESSENTIAL TO EXPLANATIONS AS TO WHY CERTAIN SLOPE FIELDS DO NOT FIT CERTAIN EQUATIONS!!!!!!! YUHHHHHH!@@#$%^&*()
look at 7.9 logistic differential equations for matching
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look at page one of 7.5 for the slope field and its explanations
YUHHHH*****
11. consider the differential equation dy/dx = 6 - 2y. let y = f(x) be the particular solution to the differential equation with the initial condition f(0) = 4 a. write an equation for the line tangent to the graph of y = f(x) at x = 0. use the tangent line to approximate f(0.6) b. find the value of d^2y/dx^2 at the point (0,4). is the graph of y = f(x) concave up or concave down at the point (0,4)? give a reason for your answer. c. find y = f(x), the particular solution to the differential equation with the initial condition f(0)=4. d. for the particular solution y = f(x) found in part c, find lim x--^∞ f(x).
a. y - 4 = -2x; y = 2.8 b. concave up because d^2y/dx^2 > 0 at (0,4) c. y = e^(-2x) + 3 d. 3
9. consider the differential equation dy/dx = e^(y)(4x-1). let y = f(x) be the particular solution to the differential equation that passes through (2,0). a. write an equation for the line tangent to the graph of f at the point (2,0). use the tangent line to approximate f(2.2). b. find y = f(x), the particular solution to the differential equation that passes through (2,0)
a. y = 7(x-2); y = 1.4 b. y = -ln(-2x^2 + x + 7)
1. determine whether the function is a solution to the differential equation y" - y = 0 a. y = sinx b. y = 4e^(-x) c. y = Ce^x
b and c
recall: d/dxsinx = d/dxcosx = S sinxdx = S cosxdx =
cosx -sinx -cosx + c sinx + c
a logistic differential equation is an equation of the form:
dy/dt = ky(1 - y/L) k is a positive constant which models the rate of growth L is a positive constant that models the limit of the growth (carrying capacity) this model is used to model populations with a carrying capacity (the maximum size populations that a given geographic region can support)
implicit differentiation: x^2 + y^2 = 9
dy/dx = x/y
12. the graph of the derivative of f, f', is shown to the right (y = Ix + 2I - 1). the domain of f is the set of all x such that -4<x<0. given that f(-2) = 0, find the solution f(x).
f(x) = {-1/2x^2 - 3x - 4, -4<x<-2; 1/2x^2 + x, -2<x<0
particular solutions
general solution but find what C is
notes on major classes of differential equations: class: constant, linear, quadratic, circle, ellipse, hyperbola, exponential, exponential vertical shift, logistic, logistic differential equation: dy/dx = 0, dy/dx = k, dy/dx = kx, dy/dx = kx - b, dy/dx = -x/y, dy/dx = -ax/by, dy/dx = ax/by, dy/dx = ky, dy/dx = k(y - L), dy/dx = (k/L)y(L - y), dy/dx = ky(1 - y/L) general solution: comments:
general solution: y=c, y=kx + c (y=mx+b), y=kx^2/2 + c, y = kx^2/2 - bx + c, x^2 + y^2 = c, ax^2 + by^2 = c, ax^2 - by^2 = c or by^2 - ax^2 = c, y = ce^(kx), y = ce^(kx) + L, y = L/(1 + be^(-kt)), y = L/(1+be^(-kt)) comments: none, increasing if k>0 and decreasing if k<0, concave up if k>0 and concave down if k<0 and c indicates vertical shift, b/k indicates horizontal shift, c= r^2, none, none, increasing if k>0 and decreasing if k<0 and c is a stretch or flip factor, L is the horizontal asymptote and c = y(0) - L, L is the horizontal asymptote and b = L/y(0) - 1, L is the horizontal asymptote and b = L/y(0) - 1
13. mr. bean's favorite addiction is put into a cylindrical container with radius 3 inches, as shown in the figure above. let h be the depth of the soda in the container, measured in inches, where h is a function of time t, measured in minutes. the volume V of soda in the container is changing at the rate of -π/2√h cubic inches per minute throughout the morning. given that h = 9 at the start of 1st period (t = 0), solve the differential equation dh/dt for h as a function of t. (the volume V of a cylinder with radius r and height h is V = πr^2h)
h = (3 - 1/36t^2)
inversely proportional (sometimes we say proportional to the reciprocal):
if a is inversely proportional to b, then a = k/b, where k is a constant
directly proportional (usually we just say proportional):
if a is proportional to b, then a = kb, where k is a constant
a population y grows according to the equation dy/dt = ky, where k is a constant and t is measured in years. if the population doubles every 7 years, what is the value of k?
k = 0.099
to find the equation of the slope field,...
look at the slopes at each point is see which equation fits when you plug the coordinate points in to get that slope
a slope field (sometimes called a direction field) is a...
map of the rate of change of a function for different values of x and y. think of the slope field as a visual that shows the general shape of all solutions to a differential equation. slope fields allow us to approximate solutions to a differential equation that is difficult or impossible to solve analytically
look at 7.5 of differential equations 7.1 to 7.5 for Euler's method
multiply dy/dx (or f') by h to get triangle y add old y(sub)n and triangle y to get new y(sub)n plug into equation to get f'
let find the approximate solution of dy/dx = x + y at x = 0, 01, 0.2, 0.3, 0.4, and 0.5 if the graph passes through the origin n: x(sub)n: y(sub)n: f'(x(sub)n, y(sub)n): (triangle, which means change in)y:
n: 0, 1, 2, 3, 4, 5 x(sub)n: 0, 0.1, 0.2, 0.3, 0.4, 0.5 y(sub)n: 0, 0, 0.01, 0.031, 0.0641, 0.11051 f'(x(sub)n, y(sub)n): 0, 0.1, 0.21, 0.331, 0.4641, 0.61051 (triangle, which means change in)y: 0, 0.01, 0.021, 0.0331, 0.04641, 0.061051
use variables and given info to find...
other variables
a slope field represents a differential equation on an xy-plane. it shows the "slope" of all the...
particular solutions to the differential equations
to find the slope on a slope field,...
plug in the coordinates to the differential equation and the answer you get is the slope at that point
a growth model will have a... a decay model will have a...
positive exponent negative exponent
the rate of change of a quantity is proportional to the...
size of the quantity. this represents an exponential model
derivatives can be used to verify that a function is a...
solution to a given differential equation
10. one of mr. Kelly's calculus students attempted to solve the differential equation dy/dx = 2xy with initial condition y = 3 when x = 0. in which step, if any, does an error first appear? step 1: S (1/y)dy = S 2xdx step 2: lnIyI = x^2 + c step 3: IyI = e^(x^2) + c step 4: since y = 3 when x = 0, 3 = e^0 + c step 5: y = -e^(x^2) + 2
step 3
general solution
take antiderivative and add C
Euler's method uses a recursive formula
x(subscript)(n+1) = x(sub)n + h y(sub)(n+1) = y(sub)n + hf'(x(sub)n, y(sub)n) h is a little number
let f be the function that satisfies the given differential equation (dy/dx = xy/2). write an equation for the tangent line to the curve y = f(x) through the point (1,1). then use your tangent line equation to estimate the value of f(1.2).
y - 1 = 1/2(x-1) y = 1.1
scientists use the parameter b instead of c to arrive at the general solution to the logistic differential equation
y = L/(1 + be^(-kt))
differentiate: dy/dt = ky(1 - y/L)
y = L/(1 + ce^(-kt))
exponential growth and decay
y = a(b)^t a = initial value b = growth/decay factor
look at all of 73. for slope fields!!!!!!!!!
yuh!!!!
look at slope fields on page 3 and 4 of 7.5
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slope fields on page 3 of differential equations 7.1 to 7.5
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