State the two key equations giving the relationships between ex and ln x.
> Find a formula for d/dx f (g (x)), where f (x) is a function such that f (x) = 1/(x2 + 1). g (x) = 1/x
> Find a formula for d/dx f (g (x)), where f (x) is a function such that f (x) = 1/(x2 + 1). g (x) = x3
> A company pays y dollars in taxes when its annual profit is P dollars. If y is some (differentiable) function of P and P is some function of time t, give a chain rule formula for the time rate of change of taxes dy/dt.
> A store estimates that its cost when selling x lamps per day is C dollars, where C = 40x + 30 (the marginal cost per lamp is $40). If daily sales are rising at the rate of three lamps per day, how fast are the costs rising? Explain your answer using the
> Repeat Exercise 17, with the sidewalk on the inside of all four sides. In this case, the 800-square-meter planted region has dimensions x - 4 meters by y - 4 meters. Exercise 17: A botanical display is to be constructed as a rectangular region with a ri
> A botanical display is to be constructed as a rectangular region with a river as one side and a sidewalk 2 meters wide along the inside edges of the other three sides. (See Fig. 1.) The area for the plants must be 800 square meters. Find the outside dime
> Find the equation of the line tangent to the graph of y = x – 3/√(4 + x2) at the point where x = 0.
> Find the equation of the line tangent to the graph of y = (x3 - 1)(x2 + 1)4 at the point where x = -1.
> Let f (x) = (x2 + 1)/(x2 + 5). Find all x such that f ‘ (x) = 0.
> Given f (1) = 1, f ‘(1) = 5, g (1) = 3, g ’(1) = 4, f ‘(3) = 2, and g ‘(3) = 6, compute. d/dx [ f ( f (x))] |x=1
> The derivative of (x3 - 4x)>x is obviously 2x for x ≠ 0, because (x3 - 4x)/x = x2 ≠ 4 for x 0. Verify that the quotient rule gives the same derivative.
> Let f (x) = (3x + 1)4(3 - x)5. Find all x such that f ‘(x) = 0.
> Differentiate the following functions. y = (x3 + x)/(x2 – x)
> Differentiate the following functions. y = [(3 - x2)/x3]2
> Differentiate the following functions. y = 2x/(2 - 3x)
> Differentiate the following functions. y = (x2 - 6x)/(x – 2)
> Differentiate the following functions. y = 1/(x2 + 5x + 1)6
> Differentiate the following functions. y = 3(x2 - 1)3(x2 + 1)5
> Differentiate the following functions. y = √x/(√x + 4)
> What does it mean for a function to be defined implicitly by an equation?
> Given f (1) = 1, f ‘(1) = 5, g (1) = 3, g ’(1) = 4, f ‘(3) = 2, and g ‘(3) = 6, compute. d/dx [ g ( f (x))] |x=1
> What is the relationship between the chain rule and the general power rule?
> Let f (x) = 1/x and g (x) = x3. (a) Show that the product rule yields the correct derivative of (1/x)x3 = x2. (b) Compute the product f (x)g (x), and note that it is not the derivative of f (x)g (x).
> State the chain rule. Give an example.
> State the product rule and the quotient rule.
> Describe an application of the differential equation y ’ = ky(M - y).
> Describe an application of the differential equation y ’ = k(M - y).
> Define the elasticity of demand, E(p), for a demand function. How is E(p) used?
> What is the difference between a relative rate of change and a percentage rate of change?
> State the formula for each of the following quantities: (a) The compound amount of P dollars in t years at interest rate r, compounded continuously (b) The present value of A dollars in n years at interest rate r, compounded continuously
> Explain how radiocarbon dating works.
> Given f (1) = 1, f ‘(1) = 5, g (1) = 3, g ’(1) = 4, f ‘(3) = 2, and g ‘(3) = 6, compute. d/dx [ f (g (x))] |x=1
> What is meant by the half-life of a radioactive element?
> What is a growth constant? A decay constant?
> 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(g (x) - f (x))] |x=2
> What differential equation is key to solving exponential growth and decay problems? State a result about the solution to this differential equation.
> What is a logarithm?
> What are the coordinates of the reflection of the point (a, b) across the line y = x?
> State the properties that graphs of the form y = ekx have in common when k is positive and when k is negative.
> Write the differential equation satisfied by y = Cekt.
> 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 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)]
> Differentiate. y = ln[(x + 5)(2x - 1)(4 - x)]
> 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
