Logistic Growth Suppose that the rate of change of the size of a population is given by

In Problems 18, find the derivative at the indicated point from the graph of each function. f (x) = 5; x = 1

In Problems 18, find the derivative at the indicated point from the graph of each function. f (x) = 3x; x = 2

In Problems 18, find the derivative at the indicated point from the graph of each function. f (x) = 4x 3; x = 1

In Problems 18, find the derivative at the indicated point from the graph of each function. f (x) = 5x + 1; x = 0

In Problems 18, find the derivative at the indicated point from the graph of each function. f (x) = 2x2; x = 0

In Problems 18, find the derivative at the indicated point from the graph of each function. f (x) = (x + 2)2; x = 1

In Problems 18, find the derivative at the indicated point from the graph of each function. f (x) = cos x; x = 0

In Problems 18, find the derivative at the indicated point from the graph of each function. f (x) = sin x; x = 2 c f _ (c) = 0

In Problems 916, find c so that f  (c) = 0. f (x) = 3x2 + 1

In Problems 916, find c so that f  (c) = 0. f (x) = x2 + 4

In Problems 916, find c so that f  (c) = 0. f (x) = (x 2)2

In Problems 916, find c so that f  (c) = 0. f (x) = (x + 3)2

In Problems 916, find c so that f  (c) = 0. f (x) = x2 6x + 9

In Problems 916, find c so that f  (c) = 0. f (x) = x2 + 4x + 4

In Problems 916, find c so that f  (c) = 0. f (x) = sin _ 2 x _

In Problems 916, find c so that f  (c) = 0. cos( x) f (c+h) f (c)

In Problems 1720, compute f (c+h) f (c) at the indicated point f (x) = 2x + 1; c = 2

In Problems 1720, compute f (c+h) f (c) at the indicated point f (x) = 3x2; c = 1

In Problems 1720, compute f (c+h) f (c) at the indicated point f (x) = x; c = 4

In Problems 1720, compute f (c+h) f (c) at the indicated point f (x) = 1 x ; c = 2

(a) Use the formal definition of the derivative to find the derivative of y = 5x2 at x = 1. (b) Show that the point (1, 5) is on the graph of y = 5x2, and find the equation of the tangent line at the point (1, 5). (c) Graph y = 5x2 and the tangent line at the point (1, 5) in the same coordinate system.

(a) Use the formal definition to find the derivative of y = 2x2 at x = 1. (b) Show that the point (1,2) is on the graph of y = 2x2, and find the equation of the tangent line at the point (1,2). (c) Graph y = 2x2 and the tangent line at the point (1,2) in the same coordinate system.

(a) Use the formal definition to find the derivative of y = 1 x3 at x = 2. (b) Show that the point (2,7) is on the graph of y = 1x3, and find the equation of the normal line at the point (2,7). (c) Graph y = 1 x3 and the tangent line at the point (2,7) in the same coordinate system.

(a) Use the formal definition to find the derivative of y = 1 x at x = 2. (b) Show that the point (2, 1 2 ) is on the graph of y = 1 x , and find the equation of the normal line at the point (2, 1 2 ). (c) Graph y = 1 x and the tangent line at the point (2, 1 2 ) in the same coordinate system.

Use the formal definition to find the derivative of y = x for x > 0.

Use the formal definition to find the derivative of f (x) = 1 x + 1 for x _= 1.

Find the equation of the tangent line to the curve y = 3x2 at the point (1, 3).

Find the equation of the tangent line to the curve y = 2/x at the point (2, 1).

Find the equation of the tangent line to the curve y = x at the point (4, 2).

Find the equation of the tangent line to the curve y = x2 3x + 1 at the point (2,1).

Find the equation of the normal line to the curve y = 3x2 at the point (1,3).

Find the equation of the normal line to the curve y = 4/x at the point (1,4).

Find the equation of the normal line to the curve y = 2x2 1 at the point (1, 1).

Find the equation of the normal line to the curve y = _ x 1 at the point (5, 2).

The following limit represents the derivative of a function f at the point (a, f (a)): lim h0 2(a + h)2 2a2 h Find f (x).

The following limit represents the derivative of a function f at the point (a, f (a)): lim h0 4(a + h)3 4a3 h Find f (x).

The following limit represents the derivative of a function f at the point (a, f (a)): lim h0 1 (2+h)2+1 1 5 h Find f and a.

The following limit represents the derivative of a function f at the point (a, f (a)): lim h0 sin( 6 + h) sin 6 h Find f and a.

Velocity A car moves along a straight road. Its location at time t is given by s(t) = 20t2, 0 t 2 where t is measured in hours and s(t) is measured in kilometers. (a) Graph s(t) for 0 t 2. (b) Find the average velocity of the car between t = 0 and t = 2. Illustrate the average velocity on the graph of s(t). (c) Use calculus to find the instantaneous velocity of the car at t = 1. Illustrate the instantaneous velocity on the graph of s(t).

Velocity A train moves along a straight line. Its location at time t is given by s(t) = 100 t , 1 t 5 where t is measured in hours and s(t) is measured in kilometers. (a) Graph s(t) for 1 t 5. (b) Find the average velocity of the train between t = 1 and t = 5. Where on the graph of s(t) can you find the average velocity? (c) Use calculus to find the instantaneous velocity of the train at t = 2. Where on the graph of s(t) can you find the instantaneous velocity?What is the speed of the train at t = 2?

Velocity If s(t) denotes the position of an object that moves along a straight line, then _s/_t, called the average velocity, is the average rate of change of s(t), and v(t) = ds/dt, called the (instantaneous) velocity, is the instantaneous rate of change of s(t). The speed of the object is the absolute value of the velocity, |v(t)|. Suppose now that a car moves along a straight road. The location at time t is given by s(t) = 160 3 t2, 0 t 1 where t is measured in hours and s(t) is measured in kilometers. (a) Where is the car at t = 3/4, and where is it at t = 1? (b) Find the average velocity of the car between t = 3/4 and t = 1. (c) Find the velocity and the speed of the car at t = 3/4.

Velocity Suppose a particle moves along a straight line. The position at time t is given by s(t) = 3t t2, t 0 where t is measured in seconds and s(t) is measured in meters. (a) Graph s(t) for t 0. (b) Use the graph in (a) to answer the following questions: (i) Where is the particle at time 0? (ii) Is there another time at which the particle visits the location where it was at time 0? (iii) How far to the right on the straight line does the particle travel? (iv) How far to the left on the straight line does the particle travel? (v) Where is the velocity positive? where negative? equal to 0? (c) Find the velocity of the particle. (d) When is the velocity of the particle equal to 1 m/s?

Tilmans Resource Model In Subsection 4.1.2, we considered Tilmans resource model. Denote the biomass at time t by B(t), and assume that 1 B dB dt = f (R) m where R denotes the resource level, f (R) = 200 R 5 + R and m = 40. Use the graphical approach to find the value R at which 1 B dB dt = 0. Then compute R by solving 1 B dB dt = 0.

Exponential Growth Assume that N(t) denotes the size of a population at time t and that N(t) satisfies the differential equation dN dt = rN where r is a constant. (a) Find the per capita growth rate. (b) Assume that r < 0 and that N(0) = 20. Is the population size at time 1 greater than 20 or less than 20? Explain your answer.

Logistic Growth Assume that N(t) denotes the size of a population at time t and that N(t) satisfies the differential equation dN dt = 3N _ 1 N 20 _ Let f (N) = 3N(1 N 20 ) for N 0. Graph f (N) as a function of N and identify all equilibria (i.e., all points where dN dt = 0).

Island Model Assume that a species lives in a habitat that consists of many islands close to a mainland. The species occupies both the mainland and the islands, but, although it is present on the mainland at all times, it frequently goes extinct on the islands. Islands can be recolonized by migrants from the mainland. The following model keeps track of the fraction of islands occupied: Denote the fraction of islands occupied at time t by p(t). Assume that each island experiences a constant risk of extinction and that vacant islands (the fraction 1p) are colonized from the mainland at a constant rate. Then dp dt = c(1 p) ep where c and e are positive constants. (a) The gain from colonization is f (p) = c(1 p) and the loss from extinction is g(p) = ep. Graph f (p) and g(p) for 0 p 1 in the same coordinate system. Explain why the two graphs intersect whenever e and c are both positive. Compute the point of intersection and interpret its biological meaning. (b) The parameter c measures how quickly a vacant island becomes colonized from the mainland. The closer the islands, the larger is the value of c. Use your graph in (a) to explain what happens to the point of intersection of the two lines as c increases. Interpret your result in biological terms.

Chemical Reaction Consider the chemical reaction A + B AB If x(t) denotes the concentration of AB at time t, then dx dt = k(a x)(b x) where k is a positive constant and a and b denote the concentrations of A and B, respectively, at time 0. Assume that k = 3, a = 7, and b = 4. For what values of x is dx/dt = 0? Interpret the meaning of dx/dt = 0.

Chemical Reaction Consider the autocatalytic reaction A + X X which was introduced in Problem 30 of Section 1.2. Find a differential equation that describes the rate of change of the concentration of the product X.

Logistic Growth Suppose that the rate of change of the size of a population is given by dN dt = rN _ 1 N K _ where N = N(t) denotes the size of the population at time t and r and K are positive constants. Find the equilibrium size of the populationthat is, the size at which the rate of change is equal to 0. Use your answer to explain why K is called the carrying capacity.

Biotic Diversity (Adapted fromValentine, 1985.)Walker and Valentine (1984) suggested a model for species diversity which assumes that species extinction rates are independent of diversity but speciation rates are regulated by competition. Denoting the number of species at time t by N(t), the speciation rate by b, and the extinction rate by a, they used the model dN dt = N _ b _ 1 N K _ a _ where K denotes the number of niches, or potential places for species in the ecosystem. (a) Find possible equilibria under the condition a < b. (b) Use your result in (a) to explain the following statement by Valentine (1985): In this situation, ecosystems are never full, with all potential niches occupied by species so long as the extinction rate is above zero. (c) What happens when a b?

Which of the following statements is true? (A) If f (x) is continuous, then f (x) is differentiable. (B) If f (x) is differentiable, then f (x) is continuous.

Explain the relationship between continuity and differentiability.

Sketch the graph of a function that is continuous at all points in its domain and differentiable in the domain except at one point.

Sketch the graph of a periodic function defined on R that is continuous at all points in its domain and differentiable in the domain except at c = k, k Z.

If f (x) is differentiable for all x R except at x = c, is it true that f (x) must be continuous at x = c? Justify your answer.

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) y = |x 2|

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) y = |x + 5|

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) y = 2 |x 3|

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) y = |x + 2| 1

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) y = 1 2 + x

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) y = 1 x 3

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) y = 3 x 3 + x

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) y = x 1 x + 1

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) y = |x2 3|

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) y = |2x2 1|

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) f (x) = _ x for x 0 x + 1 for x > 0

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) f (x) = _ 2x for x 1 x + 2 for x > 1

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) f (x) = _ x2 for x 1 2 x2 for x > 1

In Problems 5669, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) f (x) = _ x2 + 1 for x 0 ex for x > 0

Suppose the function f (x) is piecewise defined; that is, f (x) = f1(x) for x a and f (x) = f2(x) for x > a. Assume that f1(x) is continuous and differentiable for x < a and that f2(x) is continuous and differentiable for x > a. Sketch graphs of f (x) for the following three cases: (a) f (x) is continuous and differentiable at x = a. (b) f (x) is continuous, but not differentiable, at x = a. (c) f (x) is neither continuous nor differentiable at x = a.