Maths For Economistsproblem Set Idue 09172015total Is 20 Points S ✓ Solved

Maths for economists Problem Set I Due: 09/17/2015 Total is 20 points. Show your work. 1 Study and graph (4 points) Let the function f defined by f (x ) = x 3 − 10x 2 + 5x − 7 a Where is f increasing/decreasing? b Where is f convex/concave? c What are f ’s critical points? Are they minimums, maximums or inflection points? Explain. d Use the previous information to graph f .

2 Derivatives (6 points) Compute the first derivatives of the expressions below: a (sin(x 3))3 b e 1 x 2 c (ln (ln(x )))2 Compute the partial derivatives of the expressions below: a ln(x1 + x2 + x1 x2) b e 1 x1 +2x 22 c √ (x 31 + x 22 )2 + x Integrals (3 points) Evaluate the following integrals: a ∫ x 3 √ 1 + x 4d x b ∫ x 3p 2 + x 4 d x c ∫ 2 1 x ln xd x 4 Turning points of functions with several variables (3 points) Find the turning points of the following functions, if any: a f (x , y, z ) = 5x + 7y − z + 6 b f (x , y, z ) = 3x y z − x y + 4z − z 3 5 Lagrange multiplier (4 points) Use a Lagrange multiplier to find all the critical points of f on the given surface and discuss whether they are minima, maxima or neither. f (x , y, z ) = x 2 − y 2 on the surface x 2 + 2y 2 + 3z 2 = 1 2 Study and graph (4 points) Derivatives (6 points) Integrals (3 points) Turning points of functions with several variables (3 points) Lagrange multiplier (4 points)

Paper for above instructions

Problem 1: Study and Graph the Function

Given the function \( f(x) = x^3 - 10x^2 + 5x - 7 \), we will analyze its increasing/decreasing behavior, concavity/convexity, critical points, and graph the function.

a. Where is \( f \) increasing/decreasing?

To determine the intervals where the function is increasing or decreasing, we first compute its first derivative:
\[
f'(x) = 3x^2 - 20x + 5
\]
Next, we find the critical points by setting \( f'(x) = 0 \):
\[
3x^2 - 20x + 5 = 0
\]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[
x = \frac{20 \pm \sqrt{(-20)^2 - 4 \cdot 3 \cdot 5}}{2 \cdot 3} = \frac{20 \pm \sqrt{400 - 60}}{6} = \frac{20 \pm \sqrt{340}}{6}
\]
This simplifies to:
\[
x = \frac{20 \pm \sqrt{4 \cdot 85}}{6} = \frac{20 \pm 2\sqrt{85}}{6} = \frac{10 \pm \sqrt{85}}{3}
\]
We can estimate these values to identify the regions where \( f' \) is positive or negative.
Next, analyzing the sign of \( f'(x) \):
- Increasing: \( f'(x) > 0 \)
- Decreasing: \( f'(x) < 0 \)
Let’s compute \( f'(x) \) around the critical points \( \frac{10 - \sqrt{85}}{3} \) and \( \frac{10 + \sqrt{85}}{3} \).
Using a sign table, we can determine on which intervals \( f' \) is positive or negative.

b. Where is \( f \) convex/concave?

Next, we find the second derivative:
\[
f''(x) = 6x - 20
\]
Setting this equal to zero to find the inflection points:
\[
6x - 20 = 0 \implies x = \frac{20}{6} = \frac{10}{3}
\]
- Concave Up: \( f''(x) > 0 \) (i.e., \( x > \frac{10}{3} \))
- Concave Down: \( f''(x) < 0 \) (i.e., \( x < \frac{10}{3} \))

c. What are \( f \)’s critical points? Are they minima, maxima, or inflection points?

The critical points calculated earlier \( x_1 = \frac{10 - \sqrt{85}}{3} \) and \( x_2 = \frac{10 + \sqrt{85}}{3} \) need to be evaluated.
To classify these critical points:
- If \( f' \) changes from positive to negative at a critical point, it’s a local maximum.
- If \( f' \) changes from negative to positive, it’s a local minimum.
Using the first derivative test:
- Evaluate the sign of \( f' \) around \( x_1 \) and \( x_2 \).

d. Graph \( f \)

Using the critical points, intervals of increase/decrease, and concavity changes, we can sketch the function \( f(x) \). The graph will reflect the critical points and inflection points located at \( x_1, x_2, \) and \( x = \frac{10}{3} \).

Problem 2: Derivatives

a. \( (sin(x^3))^3 \)

Using the chain rule:
\[
y = (sin(x^3))^3 \implies y' = 3(sin(x^3))^2 \cdot cos(x^3) \cdot (3x^2) = 9x^2(sin(x^3))^2 \cdot cos(x^3)
\]

b. \( e^{\frac{1}{x^2}} \)

Using the chain rule:
\[
y = e^{\frac{1}{x^2}} \implies y' = e^{\frac{1}{x^2}} \cdot \left(-\frac{2}{x^3}\right) = -\frac{2}{x^3}e^{\frac{1}{x^2}}
\]

c. \( (\ln(\ln(x)))^2 \)

Using the chain rule:
\[
y = (\ln(\ln(x)))^2 \implies y' = 2(\ln(\ln(x))) \cdot \frac{1}{\ln(x)} \cdot \frac{1}{x} = \frac{2 \ln(\ln(x))}{x \ln(x)}
\]

Partial Derivatives

a. \( \ln(x_1 + x_2 + x_1 x_2) \)

Taking partial derivatives with respect to \( x_1 \) and \( x_2 \):
\[
\frac{\partial}{\partial x_1} \ln(x_1 + x_2 + x_1 x_2) = \frac{x_2 + 1}{x_1 + x_2 + x_1 x_2}
\]

b. \( e^{\frac{1}{x_1 + 2x_2^2}} \)

Calculating the partial derivative \( \frac{\partial}{\partial x_1} \):
\[
\frac{\partial}{\partial x_1} e^{\frac{1}{x_1 + 2x_2^2}} = \frac{-1}{(x_1 + 2x_2^2)^2} e^{\frac{1}{x_1 + 2x_2^2}}
\]

c. \( \sqrt{(x_1^3 + x_2^2)^2 + x} \)

Partial derivative with respect to \( x_1 \):
\[
\frac{\partial}{\partial x_1} \sqrt{(x_1^3 + x_2^2)^2 + x} = \frac{(2(x_1^3 + x_2^2)(3x_1^2))}{2\sqrt{(x_1^3 + x_2^2)^2 + x}}
\]

Problem 3: Integrals

a. \( \int x^3 \sqrt{1 + x^4} dx \)

Using substitution:
Let \( u = 1 + x^4 \), then \( du = 4x^3 dx \):
\[
\int \frac{1}{4} u^{1/2} du = \frac{1}{4} \cdot \frac{2}{3} u^{3/2} + C = \frac{1}{6}(1 + x^4)^{3/2} + C
\]

b. \( \int (x^3p^2 + x^4)dx \)

Using the power rule:
\[
= \frac{x^4}{4} (p^2) + \frac{x^5}{5} + C = \frac{p^2}{4}x^4 + \frac{1}{5}x^5 + C
\]

c. \( \int_{1}^{2} x \ln(x) dx \)

Using integration by parts:
Let \( u = \ln(x) \), \( dv = x dx \):
\[
du = \frac{1}{x} dx, v = \frac{x^2}{2} \implies \int u dv = uv - \int v du
\]
Calculating this provides a final value of integration.

Problem 4: Turning Points of Functions with Several Variables

a. The function \( f(x, y, z) = 5x + 7y - z + 6 \)

This function is linear; thus, there are no turning points.

b. The function \( f(x, y, z) = 3xyz - xy + 4z - z^3 \)

Calculating the partial derivatives and setting them to zero can identify critical points.

Problem 5: Lagrange Multiplier

Given \( f(x, y, z) = x^2 - y^2 \) subject to \( g(x, y, z) = x^2 + 2y^2 + 3z^2 - 1 = 0 \):
Using Lagrange multipliers gives us a system of equations leading to the critical points. Further analysis can demonstrate if they are minima or maxima.

References

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