Radioactive Decay Suppose W(t) denotes the amount of a radioactive material left after

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = e3x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = e2x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 4e13x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 3e25x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = e2x2+3x1

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = e4x22x+1

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = e7(x2+1)2

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = e3(x31)4

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = xex

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 2xe3x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = x2ex

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = (3x2 1)e1x2

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 1+ex 1+x2

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = xex 1+xex

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = ex+ex 2+ex

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = x ex+ex

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = esin(3x)

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = ecos(4x)

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = esin(x21)

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = ecos(12x3)

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = sin(ex )

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = cos(ex )

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = sin(e2x + x)

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = cos(3x ex21)

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = exp[x sin x]

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = exp[x2 2 cos x]

Differentiate the functions in Problems 152 with respect to the independent variable g(s) = exp[sec s2]

Differentiate the functions in Problems 152 with respect to the independent variable g(s) = exp[tan s3]

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = ex sin x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = e1x cos x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 3ex2+tan x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 2ex sec(3x)

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 2x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 3x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 2x+1

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 3x1

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 5 2x1

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 3 13x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 2x2+1

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 3x31

Differentiate the functions in Problems 152 with respect to the independent variable h(t) = 2t21

Differentiate the functions in Problems 152 with respect to the independent variable h(t) = 42t3t

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 2 x

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 3 x+1

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 2 x21

Differentiate the functions in Problems 152 with respect to the independent variable f (x) = 4 12x3

Differentiate the functions in Problems 152 with respect to the independent variable h(t) = 5 t

Differentiate the functions in Problems 152 with respect to the independent variable h(t) = 6 6t66

Differentiate the functions in Problems 152 with respect to the independent variable g(x) = 22 cos x

Differentiate the functions in Problems 152 with respect to the independent variable g(r ) = 23 sin r

Differentiate the functions in Problems 152 with respect to the independent variable g(r ) = 3r 1/5

Differentiate the functions in Problems 152 with respect to the independent variable g(r ) = 4r 1/4

Compute the limits in Problems 5356. lim h0 e2h 1 h

Compute the limits in Problems 5356. lim h0 e5h 1 3h

Compute the limits in Problems 5356. lim h0 eh 1 h

Compute the limits in Problems 5356. lim h0 2h 1 h

Find the length of the subtangent to the curve y = 2x at the point (1, 2).

Find the length of the subtangent to the curve y = exp[x2] at the point (2, e4) .

Population Growth Suppose that the population size at time t is N(t) = e2t , t 0 (a) What is the population size at time 0? (b) Show that dN dt = 2N

Population Growth Suppose that the population size at time t is N(t) = N0ert , t 0 where N0 is a positive constant and r is a real number. (a) What is the population size at time 0? (b) Show that dN dt = rN

Bacterial Growth Suppose that a bacterial colony grows in such a way that at time t the population size is N(t) = N(0)2t where N(0) is the population size at time 0. Find the rate of growth dN/dt. Express your solution in terms of N(t). Show that the growth rate of the population is proportional to the population size.

Bacterial Growth Suppose that a bacterial colony grows in such a way that at time t the population size is N(t) = N(0)2t where N(0) is the population size at time 0. Find the per capita growth rate.

Logistic Growth (a) Find the derivative of the logistic growth curve (see Example 3 in Section 3.3) N(t) = K 1 + _ K N(0) 1 _ ert where r and K are positive constants and N(0) is the population size at time 0. (b) Show that N(t) satisfies the equation dN dt = rN _ 1 N K _ [: Use the function N(t) given in (a) for the right-hand side, and simplify until you obtain the derivative of N(t) that you computed in (a).] (c) Plot the per capita rate of growth 1 N dN dt as a function of N, and note that it decreases with increasing population size.

Fish Recruitment Model The following model is used in the fisheries literature to describe the recruitment of fish as a function of the size of the parent stock: If we denote the number of recruits by R and the size of the parent stock by P, then R(P) = Pe P , P 0 where and are positive constants. (a) Sketch the graph of the function R(P) when = 1 and = 2. (b) Differentiate R(P) with respect to P. (c) Find all the points on the curve that have a horizontal tangent.

Von Bertalanffy Growth Model The growth of fish can be described by the von Bertalanffy growth function L(x) = L (L L0)ekx where x denotes the age of the fish and k, L, and L0 are positive constants. (a) Set L0 = 1 and L = 10. Graph L(x) for k = 1.0 and k = 0.1. (b) Interpret L and L0. (c) Compare the graphs for k = 0.1 and k = 1.0. According to which graph do fish reach L = 5 more quickly? (d) Show that d dx L(x) = k(L L(x)) That is, dL/dx L L. What does this proportionality say about how the rate of growth changes with age? (e) The constant k is the proportionality constant in (d). What does the value of k tell you about how quickly a fish grows?

Radioactive Decay Suppose W(t) denotes the amount of a radioactive material left after time t (measured in days). Assume that the radioactive decay rate of the material is 0.2/day. Find the differential equation for the radioactive decay function W(t).

Radioactive Decay Suppose W(t) denotes the amount of a radioactive material left after time t (measured in days). Assume that the radioactive decay rate of the material is 4/day. Find the differential equation for the radioactive decay function W(t).

Radioactive Decay Suppose W(t) denotes the amount of a radioactive material left after time t (measured in days). Assume that the half-life of the material is 3 days. Find the differential equation for the radioactive decay function W(t).

Radioactive Decay Suppose W(t) denotes the amount of a radioactive material left after time t (measured in days). Assume that the half-life of the material is 5 days. Find the differential equation for the radioactive decay function W(t).

Radioactive Decay Suppose W(t) denotes the amount of a radioactive material left after time t. Assume that W(0) = 15 and that dW dt = 2W(t) (a) How much material is left at time t = 2? (b) What is the half-life of this material?

Radioactive Decay Suppose W(t) denotes the amount of a radioactive material left after time t. Assume that W(0) = 6 and that dW dt = 3W(t) (a) How much material is left at time t = 4? (b) What is the half-life of the material?

Radioactive Decay Suppose W(t) denotes the amount of a radioactive material left after time t. Assume that W(0) = 10 and W(1) = 8. (a) Find the differential equation that describes this situation. (b) How much material is left at time t = 5? (c) What is the half-life of the material?

Radioactive Decay Suppose W(t) denotes the amount of a radioactive material left after time t. Assume that W(0) = 5 and W(1) = 2. (a) Find the differential equation that describes this situation. (b) How much material is left at time t = 3? (c) What is the half-life of the material?