Exam Review. Exam Review
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- Darlene Chase
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1
2 Chain Rule
3 Chain Rule d dx g(f (x)) = g (f (x))f (x)
4 Chain Rule d dx g(f (x)) = g (f (x))f (x) Write all roots as powers
5 Chain Rule d dx g(f (x)) = g (f (x))f (x) Write all roots as powers ( d dx ) x 2
6 Marginal Functions in Economics
7 Marginal Functions in Economics Marginal = Taking derivatives
8 Marginal Functions in Economics Marginal = Taking derivatives Cost C(x) (usually given)
9 Marginal Functions in Economics Marginal = Taking derivatives Cost C(x) (usually given) Average cost C(x) C(x) = C(x), x
10 Marginal Functions in Economics Marginal = Taking derivatives Cost C(x) (usually given) Average cost C(x) C(x) = C(x), x Marginal Avergage Cost
11 Marginal Functions in Economics Marginal = Taking derivatives Cost C(x) (usually given) Average cost C(x) C(x) = C(x), x Marginal Avergage Cost d dx C(x) = d dx ( ) C(x) x
12 Marginal Functions in Economics
13 Marginal Functions in Economics Revenue: If the price p is modeled as a function of the demand x (p = p(x)) then the revenue is given by
14 Marginal Functions in Economics Revenue: If the price p is modeled as a function of the demand x (p = p(x)) then the revenue is given by R(x) = xp(x)
15 Marginal Functions in Economics Revenue: If the price p is modeled as a function of the demand x (p = p(x)) then the revenue is given by R(x) = xp(x) If on the other hand the demand is a function of the price p, say f (p) then
16 Marginal Functions in Economics Revenue: If the price p is modeled as a function of the demand x (p = p(x)) then the revenue is given by R(x) = xp(x) If on the other hand the demand is a function of the price p, say f (p) then R(p) = pf (p)
17 Marginal Functions in Economics Revenue: If the price p is modeled as a function of the demand x (p = p(x)) then the revenue is given by R(x) = xp(x) If on the other hand the demand is a function of the price p, say f (p) then R(p) = pf (p) Marginal revenue: derivative of the revenue R as a function either of p or of x.
18 Marginal Functions in Economics
19 Marginal Functions in Economics Profit
20 Marginal Functions in Economics Profit P(x) = R(x) C(x)
21 Marginal Functions in Economics Profit P(x) = R(x) C(x) Marginal Profit
22 Marginal Functions in Economics Profit P(x) = R(x) C(x) Marginal Profit P (x) = R (x) C (x)
23 Marginal Functions in Economics
24 Marginal Functions in Economics Elasticity of demand
25 Marginal Functions in Economics Elasticity of demand p price
26 Marginal Functions in Economics Elasticity of demand p price x demand (as a function of the price)
27 Marginal Functions in Economics Elasticity of demand p price x demand (as a function of the price) If x = x(p) = f (p) then the elasticity of demand is defined as
28 Marginal Functions in Economics Elasticity of demand p price x demand (as a function of the price) If x = x(p) = f (p) then the elasticity of demand is defined as E(p) = pf (p) f (p)
29 Marginal Functions in Economics Elasticity of demand p price x demand (as a function of the price) If x = x(p) = f (p) then the elasticity of demand is defined as or E(p) = pf (p) f (p) E(p) = px (p) x(p)
30 Example
31 Example Consider the following model describing price as a function of the demand p = x
32 Example Consider the following model describing price as a function of the demand p = x Elasticity of demand when p = 2 (explain the meaning of the result).
33 Example Consider the following model describing price as a function of the demand p = x Elasticity of demand when p = 2 (explain the meaning of the result). Suppose that now
34 Example Consider the following model describing price as a function of the demand p = x Elasticity of demand when p = 2 (explain the meaning of the result). Suppose that now p = 0.01x 2 0.2x + 8
35 Example Consider the following model describing price as a function of the demand p = x Elasticity of demand when p = 2 (explain the meaning of the result). Suppose that now p = 0.01x 2 0.2x + 8 Elasticity of demand when x = 15 (explain the meaning of the result).
36 Higher order derivatives
37 Implicit Differentiation and related rates
38 Implicit Differentiation and related rates
39 Implicit Differentiation and related rates Equation of the tangent line to the curve x 2 y 3 y 2 + xy 1 = 0
40 Implicit Differentiation and related rates Equation of the tangent line to the curve at the point (1, 1). x 2 y 3 y 2 + xy 1 = 0
41 Implicit Differentiation and related rates Equation of the tangent line to the curve at the point (1, 1). x 2 y 3 y 2 + xy 1 = 0 The pressure P and volume V of an ideal gas (assuming that there is no heat transfer is given by)
42 Implicit Differentiation and related rates Equation of the tangent line to the curve at the point (1, 1). x 2 y 3 y 2 + xy 1 = 0 The pressure P and volume V of an ideal gas (assuming that there is no heat transfer is given by) P 5 V 7 = C,
43 Implicit Differentiation and related rates Equation of the tangent line to the curve at the point (1, 1). x 2 y 3 y 2 + xy 1 = 0 The pressure P and volume V of an ideal gas (assuming that there is no heat transfer is given by) P 5 V 7 = C, where C is a constant. Suppose that at a certain instant of time
44 Implicit Differentiation and related rates Equation of the tangent line to the curve at the point (1, 1). x 2 y 3 y 2 + xy 1 = 0 The pressure P and volume V of an ideal gas (assuming that there is no heat transfer is given by) P 5 V 7 = C, where C is a constant. Suppose that at a certain instant of time (a) The volume of the gas is 4 L,
45 Implicit Differentiation and related rates Equation of the tangent line to the curve at the point (1, 1). x 2 y 3 y 2 + xy 1 = 0 The pressure P and volume V of an ideal gas (assuming that there is no heat transfer is given by) P 5 V 7 = C, where C is a constant. Suppose that at a certain instant of time (a) The volume of the gas is 4 L, (b) The pressure is decreasing at a rate of 5kPa/sec.
46 Implicit Differentiation and related rates Equation of the tangent line to the curve at the point (1, 1). x 2 y 3 y 2 + xy 1 = 0 The pressure P and volume V of an ideal gas (assuming that there is no heat transfer is given by) P 5 V 7 = C, where C is a constant. Suppose that at a certain instant of time (a) The volume of the gas is 4 L, (b) The pressure is decreasing at a rate of 5kPa/sec. Find: the rate at which the volume is changing.
47 Differentials
48 Differentials Suppose that the sales of a business can be modeled by
49 Differentials Suppose that the sales of a business can be modeled by R(x) = 1 8 x 2 + 7x + 30,
50 Differentials Suppose that the sales of a business can be modeled by R(x) = 1 8 x 2 + 7x + 30, where x represents the spending in advertisement (thousands of dollars).
51 Differentials Suppose that the sales of a business can be modeled by R(x) = 1 8 x 2 + 7x + 30, where x represents the spending in advertisement (thousands of dollars). Estimate the change in the sales if the spending in advertisement increases from $25000 to $27000.
52 Differentials Suppose that the sales of a business can be modeled by R(x) = 1 8 x 2 + 7x + 30, where x represents the spending in advertisement (thousands of dollars). Estimate the change in the sales if the spending in advertisement increases from $25000 to $ Will this result in a net gain or a net loss?
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