Differentiate. y = ln[(x + 5)(2x - 1)(4 - x)]
> What is e?
> State as many laws of exponents as you can recall.
> Differentiate the function using one or more of the differentiation rules discussed thus far. y = 2(2x - 1)5/4 (2x + 1)3/4
> Give an example of the use of logarithmic differentiation.
> State the four algebraic properties of the natural logarithm function.
> State the differentiation formula for each of the following functions (a) f (x) = ekx (b) f (x) = eg(x) (c) f (x) = ln g(x)
> If f (x) and g (x) are differentiable functions such that f (2) = f ‘(2) = 3, g (2) = 3, and g’ (2) = 1/3, compute the derivative: d/dx [x f (x)] |x=2
> Give the formula that converts a function of the form bx to an exponential function with base e.
> What is the difference between a natural logarithm and a common logarithm?
> State the two key equations giving the relationships between ex and ln x.
> State the main features of the graph of y = ln x.
> What is the x-intercept of the graph of the natural logarithm function?
> Outline the procedure for solving a related-rates problem.
> Differentiate the function using one or more of the differentiation rules discussed thus far. y = 2(x3 - 1)(3x2 + 1)4
> State the formula for d/dx yr, where y is defined implicitly as a function of x.
> Simplify the following expressions. ln 4 + ln 6 - ln 12
> Simplify the following expressions. 3 ln ½ + ln 16
> Simplify the following expressions. ½ ln 9
> If f (x) and g (x) are differentiable functions such that f (2) = f ‘(2) = 3, g (2) = 3, and g’ (2) = 1/3, compute the derivative: d/dx [(g (x))2] |x=2
> Differentiate the functions. y = x√x
> Simplify the following expressions. ln x5 - ln x3
> Simplify the following expressions. ln 5 + ln x
> Find values of k and r for which the graph of y = kxr passes through the points (2, 3) and (4, 15).
> Determine the values of h and k for which the graph of y = hekx passes through the points (1, 6) and (4, 48).
> Differentiate the function using one or more of the differentiation rules discussed thus far. y = 5x3(2 - x)4
> In the study of epidemics, we find the equation ln(1 - y) - ln y = C - rt, where y is the fraction of the population that has a specific disease at time t. Solve the equation for y in terms of t and the constants C and r.
> Substantial empirical data show that, if x and y measure the sizes of two organs of a particular animal, then x and y are related by an allometric equation of the form ln y - k ln x = ln c, where k and c are positive constants that depend only on the typ
> Use logarithmic differentiation to differentiate the following functions. f (x) = x√x
> Use logarithmic differentiation to differentiate the following functions. f (x) = xx
> Use logarithmic differentiation to differentiate the following functions. f (x) = x√3
> Use logarithmic differentiation to differentiate the following functions. f (x) = 2x
> If f (x) and g (x) are differentiable functions such that f (2) = f ‘(2) = 3, g (2) = 3, and g’ (2) = 1/3, compute the derivative: d/dx [(f (x))2] |x=2
> Use logarithmic differentiation to differentiate the following functions. f (x) = (x - 2)3(x - 3)4 / (x + 4)5
> Use logarithmic differentiation to differentiate the following functions. f (x) = (x + 1)(2x + 1)(3x + 1) / √(4x + 1)
> Use logarithmic differentiation to differentiate the following functions. f (x) = ex(3x - 4)8
> Differentiate the function using one or more of the differentiation rules discussed thus far. y = 6x2(x - 1)3
> The world’s population was 5.51 billion on January 1, 1993, and 5.88 billion on January 1, 1998. Assume that, at any time, the population grows at a rate proportional to the population at that time. In what year will the world’s population reach 7 billio
> Use logarithmic differentiation to differentiate the following functions. f (x) = (x + 1)4 (4x - 1)2
> Differentiate. y = ( ln 4x)( ln 2x)
> Differentiate. y = ln(3x + 1) ln(5x + 1)
> Differentiate. y = ln [(x + 1)4(x3 + 2)] / (x – 1)
> Differentiate. y = ln (x + 1)4 / ex-1
> Differentiate. y = ln (x + 1) / (x – 1)
> Differentiate. y = In [√xer²¹+1]
> If f (x) and g (x) are differentiable functions such that f (2) = f ‘(2) = 3, g (2) = 3, and g’ (2) = 1/3, compute the derivative: d/dx [f (x)/g (x)] |x=2
> Differentiate. y = ln[e2x(x3 + 1)(x4 + 5x)]
> Differentiate. y = ln[(1 + x)2(2 + x)3(3 + x)4]
> Differentiate the function using one or more of the differentiation rules discussed thus far. y = (x4 + x2)10
> Differentiate. y = ln[(x + 1)(2x + 1)(3x + 1)]
> Solve the given equation for x. ln[(x - 3)(x + 2)] - ln(x + 2)2 - ln 7 = 0
> Solve the given equation for x. ln(x + 1) - ln(x - 2) = 1
> Solve the given equation for x. 2( ln x)2 + ln x - 1 = 0
> Solve the given equation for x. ln √x = √ln x
> Solve the given equation for x. 3 ln x - ln 3x = 0
> Solve the given equation for x. ( ln x)2 - 1 = 0
> If f (x) and g (x) are differentiable functions such that f (2) = f ‘(2) = 3, g (2) = 3, and g’ (2) = 1/3, compute the derivative: d/dx [ f (x)g (x)] |x=2
> Solve the given equation for x. ln x2 - ln 2x + 1 = 0
> Differentiate the function. y = (ax + b) / (cx + d)
> Solve the given equation for x. ln x4 - 2 ln x = 1
> Solve the given equation for x. ln √x - 2 ln 3 = 0
> Solve the given equation for x. ln x - ln x2 + ln 3 = 0
> Which of the following is the same as ln 9x2? (a) 2 * ln 9x (b) 3x * ln 3x (c) 2 * ln 3x (d) none of these
> Which of the following is the same as ln 8x2 / ln 2x? (a) ln 4x (b) 4x (c) ln 8x2 - ln 2x (d) none of these
> Which of the following is the same as ln(9x) - ln(3x)? (a) ln 6x (b) ln(9x) / ln(3x) (c) 6 * ln(x) (d) ln 3
> Which of the following is the same as 4 ln 2x? (a) ln 8x (b) 8 ln x (c) ln 8 + ln x (d) ln 16x4
> Evaluate the expression. Use ln 2 = .69 and ln 3 = 1.1. (a) ln 100 - 2 ln 5 (b) ln 10 + ln 1/5 (c) ln √108
> Evaluate the expression. Use ln 2 = .69 and ln 3 = 1.1. (a) ln 16 (b) ln 29 (c) ln 1/√2
> Figure 6 shows the graph of y = ½ + (x2 - 2x + 1) / (x2 - 2x + 2) for 0 ≤ x ≤ 2. Find the coordinates of the minimum point. Figure 6: 김 y= + 32 - 2x + 1 - 2x + 2
> Differentiate the function using one or more of the differentiation rules discussed thus far. y = (x2 + 5)15
> Evaluate the expression. Use ln 2 = .69 and ln 3 = 1.1. (a) ln 12 (b) ln 16 (c) ln(9 * 24)
> Evaluate the expression. Use ln 2 = .69 and ln 3 = 1.1. (a) ln 4 (b) ln 6 (c) ln 54
> Which is larger, ½ ln 16 or 13 ln 27? Explain.
> Which is larger, 2 ln 5 or 3 ln 3? Explain.
> Simplify the following expressions. ½ ln xy + 3/2 ln x/y
> Simplify the following expressions. ln x - ln x2 + ln x4
> Simplify the following expressions. e ln x2+3 ln y
> Simplify the following expressions. 5 ln x – ½ ln y + 3 ln z
> Simplify the following expressions. 3/2 ln 4 - 5 ln 2
> Simplify the following expressions. e2 ln x
> The following function may be viewed as a composite function h (x) = f ( g (x)). Find f (x) and g (x). h(x) = (4x - 3)3 + 1/4x - 3
> Figure 5 shows the graph of y = 10x / (1 + .25x2) for x ≥ 0. Find the coordinates of the maximum point. Figure 5: y y = 10x 1+ 25x²
> Simplify the following expressions. ln 2 - ln x + ln 3
> Differentiate the following functions. y = (x2 ln x) / 2
> Differentiate the following functions. y = ln x / ln 3
> Differentiate the following functions. y = 3 ln x + ln 2
> Human hands covered with cotton fabrics impregnated with the insect repellent DEPA were inserted for 5 minutes into a test chamber containing 200 female mosquitoes. The function f (x) = 26.48 - 14.09 ln x gives the number of mosquito bites received when
> Find the maximum area of a rectangle in the first quadrant with one corner at the origin, an opposite corner on the graph of y = -ln x, and two sides on the coordinate axes.
> Suppose that the total revenue function for a manufacturer is R(x) = 300 ln(x + 1), so the sale of x units of a product brings in about R(x) dollars. Suppose also that the total cost of producing x units is C(x) dollars, where C(x) = 2x. Find the value o
> If the demand equation for a certain commodity is p = 45/(ln x), determine the marginal revenue function for this commodity, and compute the marginal revenue when x = 20.
> The function y = 2x2 - ln 4x (x > 0) has one minimum point. Find its first coordinate.
> The following function may be viewed as a composite function h (x) = f ( g (x)). Find f (x) and g (x). h(x) = 1/x3 - 5x2 + 1
> The graph of the function y = x2 - ln x is shown in Fig. 8. Find the coordinates of its minimum point. Figure 8: 5 4. 3 2 1 Y 0 f(x) = x² - In x 0.5 1.0 1.5
> Use the fact that at the beginning of 1998, the population of the United States was 268,924,000 people and growing at the rate of 1,856,000 people per year. At the beginning of 1998, the annual consumption of ice cream in the United States was 12,582,000
> Repeat Exercise 31 with the functions y = x + ln x and y = ln 5x. (See Fig. 7). Figure 7: Exercise 31: The graphs of y = x + ln x and y = ln 2x are shown in Fig. 6. (a) Show that both functions are increasing for x > 0. (b) Find the point of inters
> The graphs of y = x + ln x and y = ln 2x are shown in Fig. 6. (a) Show that both functions are increasing for x > 0. (b) Find the point of intersection of the graphs. Figure 6: 3 CO 2 1 -1 -2 -3 y = x + Inx 1.0 y = In 2r + 1.5 2.0
> Repeat the previous exercise with y = √x ln x. Exercise 29: Find the coordinates of the relative extreme point of y = x2 ln x, x > 0. Then, use the second derivative test to decide if the point is a relative maximum point or a relative minimum point.
> Find the coordinates of the relative extreme point of y = x2 ln x, x > 0. Then, use the second derivative test to decide if the point is a relative maximum point or a relative minimum point.
> Find the equations of the tangent lines to the graph of y = ln |x| at x = 1 and x = -1.
> Determine the domain of definition of the given function. (a) f (t) = ln(ln t) (b) f (t) = ln(ln(ln t))
> The function f (x) = (ln x + 1)>x has a relative extreme point for x > 0. Find the coordinates of the point. Is it a relative maximum point?
