Differentiate the following functions. y = (√x + 1)e-2x
> The population of a certain country is growing exponentially. The total population (in millions) in t years is given by the function P(t). Match each of the following answers with its corresponding question. Answers a. Solve P(t) = 2 for t. b. P(2) c. P
> A colony of bacteria is growing exponentially with growth constant .4, with time measured in hours. Determine the size of the colony when the colony I growing at the rate of 200,000 bacteria per hour. Determine the rate at which the colony will be growin
> Find b so that 8-x/3 = bx for all x.
> The population of a city t years after 1990 satisfies the differential equation y = .02y. What is the growth constant? How fast will the population be growing when the population reaches 3 million people? At what level of population will the population
> Two different bacteria colonies are growing near a pool of stagnant water. The first colony initially has 1000 bacteria and doubles every 21 minutes. The second colony has 710,000 bacteria and doubles every 33 minutes. How much time will elapse before th
> The atmospheric pressure P(x) (measured in inches of mercury) at height x miles above sea level satisfies the differential equation P (x) = -.2P(x). Find the formula for P(x) if the atmospheric pressure at sea level is 29.92.
> Differentiate the functions. y = (-x3 + 2) (x/2 – 1)
> The population (in millions) of a state t years after 2010 is given by the graph of the exponential function y = P(t) with growth constant .025 in Fig. 6. [In parts (c) and (d) use the differential equation satisfied by P(t).] Figure 6: (a) What is the
> Simplify the following. (ex2)3
> Calculate the following. 40.2 * 40.3
> Calculate the following. 95/2 / 93/2
> Calculate the following. 81/2 * 21/2
> Differentiate the function. y = x / ln x
> Differentiate the function. y = ln(x6 + 3x4 + 1)
> Solve the following equations for t. 2e- 0.3t = 1
> Find a number b such that the function f (x) = 3-2x can be written in the form f (x) = bx.
> Calculate the following. (25/7)14/5
> Solve the following equations for t. 2 ln t = 5
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = 1/x , g (x) = 1 - x2
> Solve the following equations for t. 3et/2 - 12 = 0
> Solve the following equations for t. 3e2t = 15
> Solve the following equations for t. ln( ln 3t) = 0
> Solve the following equations for t. t ln t = e
> Simplify the following expressions. [e ln x]2
> Simplify the following expressions. e-5 ln 1
> Simplify the following expressions. e2 ln 2
> Simplify the following expressions. ln x2 / ln x3
> Write expression in the form 2kx or 3kx, for a suitable constant k. (3-x * 3x/5)5, (161/4 * 16-3/4)3x
> Simplify the following expressions. e ln(x2)
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = √x, g (x) = x2 + 1
> Calculate the following. 16- 0.25
> Simplify the following expressions. e( ln 5)/2
> Solve the equation for t. Show that the tangent lines to the graph of y = (ex - e-x) / (ex + e-x) at x = 1 and x = -1 are parallel.
> Solve the equation for t. Find the equation of the tangent line to the graph of y = ex / (1 + ex) at (0,.5).
> Solve the equation for t. Determine the intervals where the function f (x) = x ln x (x > 0) is increasing and where it is decreasing.
> Solve the equation for t. Determine the intervals where the function f (x) = ln(x2 + 1) is increasing and where it is decreasing.
> Solve the equation for t. Find the points on the graph of y = ex + e-2x where the tangent line is horizontal.
> Solve the equation for t. Find the points on the graph of y = ex where the tangent line has slope 4.
> Solve the equation for t. Solve the equation 3x = 2ex.
> Write expression in the form 2kx or 3kx, for a suitable constant k. (2-3x * 2-2x)2/5, (91/2 * 94)x/9
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = x5, g (x) = 6x - 1
> Solve the equation for t. Solve the equation 4 * 2x = ex.
> Solve the equation for t. et - 8e0.02t = 0
> Calculate the following. 5-2
> Solve the equation for t. 4e0.03t - 2e0.06t = 0
> Show that the function in Fig. 1 has a relative maximum at x = 0 by determining the concavity of the graph at x = 0. Figure 1: f(x) = ex² - 4x² -1.5 -1-0.5 12 10 8 6 4 2 Y 0.5 1 1.5 X
> The graph of the functions f (x) = ex2 - 4x2 is shown in Fig. 1. Find the first coordinates of the relative extreme points. Figure 1: f(x) = ex² - 4x² -1.5 -1-0.5 12 10 8 6 4 2 Y 0.5 1 1.5 X
> Differentiate the following functions. y = xe
> Differentiate the following functions. y = (x2 - x + 5) / (e3x + 3)
> Differentiate the following functions. y = eex
> Sketch the graph of y = 2/(1 + x2).
> Write expression in the form 2kx or 3kx, for a suitable constant k. 25x/4 * (1/2)x, 3-2x * 35x/2
> Differentiate the following functions. y = (ex + 1)/(x – 1)
> Differentiate the following functions. y = x ex2
> Differentiate the following functions. y = e√x
> Calculate the following. 41.5
> Differentiate the following functions. y = 10e7x
> Solve the following equations for x. e-5x * e4 = e
> Solve the following equations for x. (ex * e2)3 = e-9
> Solve the following equations for x. ex2-x = e2
> Solve the following equations for x. e-3x = e-12
> Sketch the graph of y = 4x/(x + 1)2, x >-1.
> Simplify the following. (e5x/2 - e3x) √ex
> Write expression in the form 2kx or 3kx, for a suitable constant k. 23x * 2-5x/2, 32x * (1/3)2x/3
> Simplify the following. (e8x + 7e-2x) e3x
> Simplify the following. 2x * 3x
> Simplify the following. e3x ex
> Simplify the following. e5x * e2x
> Calculate the following. 274/3
> The health expenditures (in billions of dollars) for a certain country from 1990 to 2010 are given approximately by f (t) = 27e0.106t, with time in years measured from 1990. Give approximate answers to the following questions using the graphs of f (t) an
> The atmospheric pressure at an altitude of x kilometers is f (x) g/cm2 (grams per square centimeter), where f (x) = 1035e-0.12x. Give approximate answers to the following questions using the graphs of f (x) and f ‘(x) shown in Fig. 2.
> Use logarithmic differentiation to differentiate the function. f (x) = exx22x
> Function h(x) is defined in terms of a differentiable f (x). Find an expression for h (x). h(x) = √[f (x2)]
> Use logarithmic differentiation to differentiate the function. f (x) = ex+1(x2 + 1)x
> Use logarithmic differentiation to differentiate the function. f (x) = [ex √(x + 1) (x2 + 2x + 3)2]/4x2
> Write expression in the form 2kx or 3kx, for a suitable constant k. 2x/6x, 3-5x/3-2x, 16x/8-x
> Use logarithmic differentiation to differentiate the function. f (x) = [(xex)/(x3 + 3)]
> Use logarithmic differentiation to differentiate the function. f (x) = √(x2 + 5) ex2
> Use logarithmic differentiation to differentiate the function. f (x) = 10x
> Use logarithmic differentiation to differentiate the function. f (x) = x1+x
> Use logarithmic differentiation to differentiate the function. f (x) = (x2 + 5)6(x3 + 7)8(x4 + 9)10
> Use logarithmic differentiation to differentiate the function. f (x) = bx, where b > 0
> Use logarithmic differentiation to differentiate the function. f (x) = x√x
> Function h(x) is defined in terms of a differentiable f (x). Find an expression for h (x). h(x) = f (x2)/x
> Use logarithmic differentiation to differentiate the function. f (x) = 2x
> Use logarithmic differentiation to differentiate the function. f (x) = 5√(x5 + 1)/(x5 + 5x + 1)
> Differentiate the function. y = ln(ex + 3e-x)
> Write expression in the form 2kx or 3kx, for a suitable constant k. 4x, (√3)x, (1/9)x
> Differentiate the functions. y = (x2 + 3)(x2 - 3)10
> Differentiate the function. y = ln (1 /e√x)
> Differentiate the function. y = e2 ln(2x+1)
> Differentiate the function. y = ln |x – 1|
> Differentiate the function. y = ln(3x+1) - ln 3
> Differentiate the function. y = ln(2x)
> Function h(x) is defined in terms of a differentiable f (x). Find an expression for h (x). h(x) = f (f (x))
> Differentiate the function. y = ln 3√(x3 + 3x – 2)
> Differentiate the function. y = ln(ex2/x)
> Differentiate the function. y = ln |-2x + 1|
> Differentiate the function. y = ln √[(x2 + 1) / (2x + 3)]
> Differentiate the function. y = ln(x2 + ex)
