In Exercises 16, investigate the one-parameter family of functions. Assume that a is

In Exercises 16, investigate the one-parameter family of functions. Assume that a is positive. (a) Graph f(x) using three different values for a. (b) Using your graph in part (a), describe the critical points of f and how they appear to move as a increases. (c) Find a formula for the x-coordinates of the critical point(s) of f in terms of a.

In Exercises 16, investigate the one-parameter family of functions. Assume that a is positive. (a) Graph f(x) using three different values for a. (b) Using your graph in part (a), describe the critical points of f and how they appear to move as a increases. (c) Find a formula for the x-coordinates of the critical point(s) of f in terms of a.

In Exercises 16, investigate the one-parameter family of functions. Assume that a is positive. (a) Graph f(x) using three different values for a. (b) Using your graph in part (a), describe the critical points of f and how they appear to move as a increases. (c) Find a formula for the x-coordinates of the critical point(s) of f in terms of a.

In Exercises 16, investigate the one-parameter family of functions. Assume that a is positive. (a) Graph f(x) using three different values for a. (b) Using your graph in part (a), describe the critical points of f and how they appear to move as a increases. (c) Find a formula for the x-coordinates of the critical point(s) of f in terms of a.

In Exercises 16, investigate the one-parameter family of functions. Assume that a is positive. (a) Graph f(x) using three different values for a. (b) Using your graph in part (a), describe the critical points of f and how they appear to move as a increases. (c) Find a formula for the x-coordinates of the critical point(s) of f in terms of a.

In Exercises 16, investigate the one-parameter family of functions. Assume that a is positive. (a) Graph f(x) using three different values for a. (b) Using your graph in part (a), describe the critical points of f and how they appear to move as a increases. (c) Find a formula for the x-coordinates of the critical point(s) of f in terms of a.

Consider the family y = A x + B . (a) If B = 0, what is the effect of varying A on the graph? (b) If A = 1, what is the effect of varying B? (c) On one set of axes, graph the function for several values of A and B.

If A and B are positive constants, find all critical points of f(w) = A w2 B w .

The graphs of f(x)=1+ eax for a = 1, 2, and 5, are in Figure 4.63. Without a calculator, identify the graphs by looking at f (0).

The graphs of f(x) = xeax for a = 1, 2 , and 3, are in Figure 4.64. Without a calculator, identify the graphs by locating the critical points of f(x).

In Exercises 1116, investigate the given two parameter family of functions. Assume that a and b are positive. (a) Graph f(x) using b = 1 and three different values for a. (b) Graph f(x) using a = 1 and three different values for b. (c) In the graphs in parts (a) and (b), how do the critical points of f appear to move as a increases? As b increases? (d) Find a formula for the x-coordinates of the critical point(s) of f in terms of a and b.

In Exercises 1116, investigate the given two parameter family of functions. Assume that a and b are positive. (a) Graph f(x) using b = 1 and three different values for a. (b) Graph f(x) using a = 1 and three different values for b. (c) In the graphs in parts (a) and (b), how do the critical points of f appear to move as a increases? As b increases? (d) Find a formula for the x-coordinates of the critical point(s) of f in terms of a and b.

In Exercises 1116, investigate the given two parameter family of functions. Assume that a and b are positive. (a) Graph f(x) using b = 1 and three different values for a. (b) Graph f(x) using a = 1 and three different values for b. (c) In the graphs in parts (a) and (b), how do the critical points of f appear to move as a increases? As b increases? (d) Find a formula for the x-coordinates of the critical point(s) of f in terms of a and b.

In Exercises 1116, investigate the given two parameter family of functions. Assume that a and b are positive. (a) Graph f(x) using b = 1 and three different values for a. (b) Graph f(x) using a = 1 and three different values for b. (c) In the graphs in parts (a) and (b), how do the critical points of f appear to move as a increases? As b increases? (d) Find a formula for the x-coordinates of the critical point(s) of f in terms of a and b.

In Exercises 1116, investigate the given two parameter family of functions. Assume that a and b are positive. (a) Graph f(x) using b = 1 and three different values for a. (b) Graph f(x) using a = 1 and three different values for b. (c) In the graphs in parts (a) and (b), how do the critical points of f appear to move as a increases? As b increases? (d) Find a formula for the x-coordinates of the critical point(s) of f in terms of a and b.

In Exercises 1116, investigate the given two parameter family of functions. Assume that a and b are positive. (a) Graph f(x) using b = 1 and three different values for a. (b) Graph f(x) using a = 1 and three different values for b. (c) In the graphs in parts (a) and (b), how do the critical points of f appear to move as a increases? As b increases? (d) Find a formula for the x-coordinates of the critical point(s) of f in terms of a and b.

The graphs of the function f(x) = x + a2/x for a = 1 and 2, and a third integer value of a, are shown in Figure 4.65. Without a calculator, identify the graphs, and estimate the third value of a.

The graphs of the function f(x) = x + a sin x for various values of a > 0 are shown in Figure 4.66. Explain why, no matter what the positive value of a, the curves intersect in the same points.

(a) Sketch graphs of y = xebx for b = 1, 2, 3, 4. Describe the graphical significance of b. (b) Find the coordinates of the critical point of y = xebx and use it to confirm your answer to part (a)

Consider the surge function y = axebx for a, b > 0. (a) Find the local maxima, local minima, and points of inflection. (b) How does varying a and b affect the shape of the graph? (c) On one set of axes, graph this function for several values of a and b

Find a formula for the family of cubic polynomials with an inflection point at the origin. How many parameters are there

(a) Derive formulas for the first and second derivatives of the logistic function: y = L 1 + Aekt for L, A, and k positive constants. (b) Derive a formula for the t value of any inflection point(s). (c) Use the second derivative to determine the concavity on either side of any inflection points.

(a) Graph f(x) = x + a sin x for a = 0.5 and a = 3. (b) For what values of a is f(x) increasing for all x?

(a) Graph f(x) = x2 + a sin x for a = 1 and a = 20. (b) For what values of a is f(x) concave up for all x?

Consider the family of functions y = a cosh(x/a) for a > 0. Sketch graphs for a = 1, 2, 3. Describe in words the effect of increasing a

Sketch several members of the family y = eax sin bx for b = 1, and describe the graphical significance of the parameter a.

Sketch several members of the family eax sin bx for a = 1, and describe the graphical significance of the parameter b.

If a > 0, b > 0, show that f(x) = a(1 ebx) is everywhere increasing and everywhere concave down.

Let f(x) = bxe1+bx, where b is constant and b > 0. (a) What is the x-coordinate of the critical point of f? (b) Is the critical point a local maximum or a local minimum? (c) Show that the y-coordinate of the critical point does not depend on the value of b

Let h(x) = ex +kx, where k is any constant. For what value(s) of k does h have (a) No critical points? (b) One critical point? (c) A horizontal asymptote

Let g(x) = x kex, where k is any constant. For what value(s) of k does the function g have a critical point?

Find formulas for the functions described in Problems 3243

Find formulas for the functions described in Problems 3243

Find formulas for the functions described in Problems 3243

Find formulas for the functions described in Problems 3243

Find formulas for the functions described in Problems 3243

Find formulas for the functions described in Problems 3243

Find formulas for the functions described in Problems 3243

Find formulas for the functions described in Problems 3243

Find formulas for the functions described in Problems 3243

Find formulas for the functions described in Problems 3243

Find formulas for the functions described in Problems 3243

Find formulas for the functions described in Problems 3243

Consider the family of functions y = f(x) = x k x, with k a positive constant and x 0. Show that the graph of f(x) has a local minimum at a point whose xcoordinate is 1/4 of the way between its x-intercepts.

For any constant a, let f(x) = ax x ln x for x > 0. (a) What is the x-intercept of the graph of f(x)? (b) Graph f(x) for a = 1 and a = 1. (c) For what values of a does f(x) have a critical point for x > 0? Find the coordinates of the critical point and decide if it is a local maximum, a local minimum, or neither.

Let f(x) = x2 + cos(kx), for k > 0. (a) Graph f for k = 0.5, 1, 3, 5. Find the smallest number k at which you see points of inflection in the graph of f. (b) Explain why the graph of f has no points of inflection if k 2, and infinitely many points of inflection if k > 2. (c) Explain why f has only a finite number of critical points, no matter what the value of k.

Let f(x) = ex kx, for k > 0. (a) Graph f for k = 1/4, 1/2, 1, 2, 4. Describe what happens as k changes. (b) Show that f has a local minimum at x = ln k. (c) Find the value of k for which the local minimum is the largest.

A family of functions is given by r(x) = 1 a + (x b)2 . (a) For what values of a and b does the graph of r have a vertical asymptote? Where are the vertical asymptotes in this case? (b) Find values of a and b so that the function r has a local maximum at the point (3, 5).

(a) Find all critical points of f(x) = x4 + ax2 + b. (b) Under what conditions on a and b does this function have exactly one critical point? What is the one critical point, and is it a local maximum, a local minimum, or neither? (c) Under what conditions on a and b does this function have exactly three critical points? What are they? Which are local maxima and which are local minima? (d) Is it ever possible for this function to have two critical points? No critical points? More than three critical points? Give an explanation in each case

Let y = Aex + Bex for any constants A, B. (a) Sketch the graph of the function for (i) A = 1, B = 1 (ii) A = 1, B = 1 (iii) A = 2, B = 1 (iv) A = 2, B = 1 (v) A = 2, B = 1 (vi) A = 2, B = 1 (b) Describe in words the general shape of the graph if A and B have the same sign. What effect does the sign of A have on the graph? (c) Describe in words the general shape of the graph if A and B have different signs. What effect does the sign of A have on the graph? (d) For what values of A and B does the function have a local maximum? A local minimum? Justify your answer using derivatives.

The temperature, T , in C, of a yam put into a 200C oven is given as a function of time, t, in minutes, by T = a(1 e kt) + b. (a) If the yam starts at 20C, find a and b. (b) If the temperature of the yam is initially increasing at 2C per minute, find k.

For positive a, b, the potential energy, U, of a particle is U = b a2 x2 a x for x > 0. (a) Find the intercepts and asymptotes. (b) Compute the local maxima and minima. (c) Sketch the graph.

The force, F, on a particle with potential energy U is given by F = dU dx . Using the expression for U in Problem 52, graph F and U on the same axes, labeling intercepts and local maxima and minima.

The Lennard-Jones model predicts the potential energy V (r) of a two-atom molecule as a function of the distance r between the atoms to be V (r) = A r12 B r6 , r> 0, where A and B are positive constants. (a) Evaluate limr0+ V (r), and interpret your answer.

For positive A, B, the force between two atoms is a function of the distance, r, between them: f(r) = A r2 + B r3 r > 0. (a) What are the zeros and asymptotes of f? (b) Find the coordinates of the critical points and inflection points of f. (c) Graph f. (d) Illustrating your answers with a sketch, describe the effect on the graph of f of: (i) Increasing B, holding A fixed (ii) Increasing A, holding B fixed

An organism has size W at time t. For positive constants A, b, and c, the Gompertz growth function gives W = Aeebct , t 0. (a) Find the intercepts and asymptotes. (b) Find the critical points and inflection points. (c) Graph W for various values of A, b, and c. (d) A certain organism grows fastest when it is about 1/3 of its final size. Would the Gompertz growth function be useful in modeling its growth? Explain.

Every function of the form f(x) = x2 + bx + c, where b and c are constants, has two zeros.

Every function of the form f(x) = a/x + bx, where a and b are non-zero constants, has two critical points.

Every function of the form f(x) = a/x + bx, where a and b are non-zero constants, has two critical points.

A member of the family f(x) = ax3 bx that has no critical points

A family of functions, f(x), depending on a parameter a, such that each member of the family has exactly one critical point.

A family of functions, g(x), depending on two parameters, a and b, such that each member of the family has exactly two critical points and one inflection point. You may want to restrict a and b.

Let f(x) = ax+b/x. Suppose a and b are positive. What happens to f(x) as b increases? (a) The critical points move further apart. (b) The critical points move closer together. (c) The critical values move further apart. (d) The critical values move closer together.

Let f(x) = ax+b/x. Suppose a and b are positive. What happens to f(x) as a increases?