**Differentiation and Integration Problems**1. **Differentiate $ y = x^2 + 5x - 5 $ using the five step rule.**2. **Differentiate:** a) $ y = x^3 - 5x + 5 $ b) $ y = xe^x $ c) $ y = \ln(x^3) $ d) $ y = e^{x}(x^2 + 2x - 1) $ (define integral) e) $ y = (x^3 + 2x + 5/x)^{12} $ f) $ y = (x^3 - 3x + 5)/x $ g) $ y = \ln(x^2 + 1)/x $3. **Find an indefinite integral and a definite integral:** a) $ y = \int \frac{(5x^4 +1)}{(x^5 + x + 7)}dx $ (indefinite integral) b) $ y = \text{Def} \int_{2}^{4} ((x^3 + 4x - 2)^2(3x^2 + 4))dx $ (definite integral)4. **If $ f(x) = e^x $, find the slope at $ x = 2 $ and the equation of the tangent line at $ x = 4 $.**5. **If $ 4x^2 + 6xy + 9y = 0 $, find the slope at the point $ (1, -1) $.**6. **If $ q = 120 - 4p $ is a demand curve:** a) Calculate the elasticity at $ p = 15 $. Is it elastic or inelastic? Where is revenue maximized? b) Calculate the revenue at $ p = 20 $. Where is the maximum revenue? c) What is the maximum revenue?7. **I sell 1000 tickets for $15 each. For each dollar I lower the price, I sell 40 more tickets. What should I charge to maximize revenue?**

Name: **Differentiation and Integration Problems**1. **Differentiate $ y =
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Description: **Differentiation and Integration Problems**1. **Differentiate \( y = x^2 + 5x - 5 $ using the five step rule.**2. **Differentiate:** a) $ y = x^3 - 5x + 5 $ b) $ y = xe^x $ c) $ y = \ln(x^3) $ d) $ y = e^{x}(x^2 + 2x - 1) $ (define

Answered: а)у-x^3-5x+5 b)y= xe^X с) In(x^3

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**Differentiation and Integration Problems**

1. **Differentiate $ y = x^2 + 5x - 5 $ using the five step rule.**

2. **Differentiate:**

a) $ y = x^3 - 5x + 5 $

b) $ y = xe^x $

c) $ y = \ln(x^3) $

d) $ y = e^{x}(x^2 + 2x - 1) $ (define integral)

e) $ y = (x^3 + 2x + 5/x)^{12} $

f) $ y = (x^3 - 3x + 5)/x $

g) $ y = \ln(x^2 + 1)/x $

3. **Find an indefinite integral and a definite integral:**

a) $ y = \int \frac{(5x^4 +1)}{(x^5 + x + 7)}dx $ (indefinite integral)

b) $ y = \text{Def} \int_{2}^{4} ((x^3 + 4x - 2)^2(3x^2 + 4))dx $ (definite integral)

4. **If $ f(x) = e^x $, find the slope at $ x = 2 $ and the equation of the tangent line at $ x = 4 $.**

5. **If $ 4x^2 + 6xy + 9y = 0 $, find the slope at the point $ (1, -1) $.**

6. **If $ q = 120 - 4p $ is a demand curve:**

a) Calculate the elasticity at $ p = 15 $. Is it elastic or inelastic? Where is revenue maximized?

b) Calculate the revenue at $ p = 20 $. Where is the maximum revenue?

c) What is the maximum revenue?

7. **I sell 1000 tickets for $15 each. For each dollar I lower the price, I sell 40 more tickets. What should I charge to maximize revenue?**

Expert Solution

Step 1

# we are entitled to solve 3 subparts of one question. Please resubmit the question which you wish to get answered.

the given equation is $y = x^{3} - 5 x + 5$

Using the formula

$\frac{d (x^{n})}{d x} = n x^{n - 1} \frac{d (a x)}{d x} = a w h e r e a i s t h e c o n s \tan t \frac{d (a)}{d x} = 0$

therefore,

$\begin{array}{rcl} \frac{d y}{d x} & = & 3 x^{2} - 5 + 0 \\ = & 3 x^{2} - 5 \end{array}$

therefore, the derivative of $y = x^{3} - 5 x + 5$ is $\begin{array}{rcl} 3 x^{2} - 5 \end{array}$

Step 2

the given equation is $y = x e^{x}$

use the product rule and formulas as shown

$\begin{array}{rcl} \frac{d (u v)}{d x} & = & u \frac{d v}{d x} + v \frac{d u}{d x} \\ \frac{d (x)}{d x} & = & 1 \\ \frac{d (e^{x})}{d x} & = & e^{x} \end{array}$

therefore,

$\begin{array}{rcl} \frac{d y}{d x} & = & x \frac{d (e^{x})}{d x} + e^{x} \frac{d x}{d x} \\ = & x e^{x} + e^{x} (1) \\ = & x e^{x} + e^{x} \end{array}$

therefore, the derivative of $y = x e^{x}$ is $\begin{array}{rcl} x e^{x} + e^{x} \end{array}$

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