第12类RD Sharma解决方案-第13章，作为速率测量工具的派生-练习13.1

问题1.求出半径r和高度h的圆柱体的总表面积在半径变化时的变化率。

解决方案：

Let total surface area of the cylinder be A 
A = 2πr(r + h) 
Now we will differentiating it with respect to r as r varies 
dA/dr = 2πr(0+1) + (h+r)2π 
dA/dr = 4πr + 2πh

问题2。求出球体体积相对于直径的变化率。

解决方案：

Let D be the diameter and r be the radius of sphere. 
So volume of sphere = 4/3πr² 
so we can write as v = 4/24πD³ [d = 2r] 
Now we will differentiating it with respect to D 
dv/dD = 12/24πD² 
dv/dD = πD²/2

问题3.求出半径为2 cm时球体体积相对于其表面积的变化率。

解决方案：

Given in question radius of sphere(r) = 2cm 
As we know that, v = 4/3πr² 
dv/dr = 4πr² —-(equation i) 

A = 4πr² 
dA/dr = 8πr² —-(equation ii) 
Dividing equation (i) and (ii) 
(dv/dr)/(dA/dr) = 4πr² / 8πr 
dv/dA = r/2 
dv/dA at r = 2 is 1.

问题4.求出半径为3 cm时圆盘面积相对于圆周的变化率。

解决方案：

Let r be the radius of circular disc. 
As we know that Area(A) = πr² 
dA/dr = 2πr —(equation i) 
circumference(C) = 2πr 
dC/dr = 2π —(equation ii) 
Dividing equation (i) by (ii) 
(dA/dr)/(dc/dr) = 2πr / 2π 
dA/dc = r 
At r = 3 dA/dc = 3.

问题5.求出圆锥体的体积相对于其底面半径的变化率。

解决方案：

Let r be the radius 
V be the volume of cone 
h be the height 
As we know that V = 1/3πr²h 
dV/dr = 2/3πrh.

问题6：求出当r = 5cm时，圆的面积相对于半径r的变化率。

解决方案：

Let r be the radius 
A be the area of circle. 
As we know that A = πr² 
dA/dr = 2πr 
At r=5 , dA/dr = 2π(5) 
= 10π

问题7：求出球的体积相对于半径r的变化率。当半径为2cm时，体积相对于半径的变化速度有多快?

解决方案：

Here given in the question , r = 2cm 
V = 4/3πr³ 
dV/dr = 4πr² 
At r = 2 , dV/dr = 4π(2)² 
= 16π

问题8.与生产一个项目的x个单位相关的以卢比为单位的总成本C(x)由C(x)= 0.007x ³ – 0.003x ² + 15x + 4000给出。找到17个单位为生产的。

解决方案：

Here in the given question: 
Marginal cost is the rate of change of total cost with respect to output. 
Marginal cost(MC) = dC/dx = 0.007(3x²) – 0.003(2x) + 15 
= 0.021x² – 0.006x + 15 
When x=17 , MC = 0.021(17²) – 0.006(17) + 15 
= 6.069 – 0.102 + 15 
=20.967 
When 17 units are produced , the marginal cost is Rs 20.967.

问题9。R(x)给出的从生产x个产品的销售中获得的以卢比为单位的总收入= 13x ² + 26x +15。找到当x = 7时的边际收入。

解决方案：

Marginal revenue is the rate of change of total revenue with respect to the number of units sold 
Marginal Revenue(MR) = dR/dx = 13(2x) + 26 = 26x + 26 
when x = 7 
MR = 26(7) + 26 = 182 +26 = 208 
So we can that required marginal cost is Rs208.

问题10.用于公司雇员福利的金钱与公司总收入(边际收入)的变化率成正比。如果从某产品的x个单位的销售中获得的总收入(以卢比为单位)由R(x)= 3x ² + 36x + 5给出，则在x = 5时找到边际收入，并写出哪个值可以解决问题表明。

解决方案：

Given function R(x) = 3x² + 36x + 5 
dR/dx = 6x + 36 
At x = 5, dR/dx = 6 x 5 + 36 
= 66 
According to the question, amount of money spent on welfare of employees.