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Saturday, February 20, 2016

VOLUME OF HEXAGONAL PRISM



DIFFERENTIATION OF EXPONENTIAL FUNCTIONS

 

DERIVATIVE OF THE EXPONENTIAL FUNCTION

The derivative of ex is quite remarkable. The expression for the derivative is the same as the expression that we started with; that is, ex!
What does this mean? It means the slope is the same as the function value (the y-value) for all points on the graph.
Example: Let's take the example when x = 2. At this point, the y-value is e2 ≈ 7.39.
Since the derivative of ex is ex, then the slope of the tangent line at x = 2 is also e2 ≈ 7.39.
We can see that it is true on the graph:

Let's now see if it is true at some other values of x.

We can see that at x = 4, the y-value is 54.6 and the slope of the tangent (in red) is also 54.6.
At x = 5, the y-value is 148.4, as is the value of the derivative and the slope of the tangent (in green).

DIFFERENTIATION OF EXPONENTIAL FUNCTIONS
The derivative of f(x) = b x is given by

f '(x) = b x ln b


Note: if f(x) = e x , then f '(x) = e x

Example 1: Find the derivative of f(x) = 2 x

Solution to Example 1:


·         Apply the formula above to obtain

f '(x) = 2 x ln 2


Example 2: Find the derivative of f(x) = 3 x + 3x 2

Solution to Example 2:


·         Let g(x) = 3 x and h(x) = 3x 2, function f is the sum of functions g and h: f(x) = g(x) + h(x). Use the sum rule, f '(x) = g '(x) + h '(x), to find the derivative of function f


f '(x) = 3 x ln 3 + 6x


Example 3: Find the derivative of f(x) = e x / ( 1 + x )

Solution to Example 3:


·         Let g(x) = e x and h(x) = 1 + x, function f is the quotient of functions g and h: f(x) = g(x) / h(x). Hence we use the quotient rule, f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x) 2, to find the derivative of function f.

g '(x) = e x

h '(x) = 1

f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x) 2

= [ (1 + x)(e x) - (e x)(1) ] / (1 + x) 2


Multiply factors in the numerator and simplify 

f '(x) = x e x / (1 + x) 2



Example 4: Find the derivative of f(x) = e 2x + 1

Solution to Example 4:


·         Let u = 2x + 1 and y = e u, Use the chain rule to find the derivative of function f as follows.

f '(x) = (dy / du) (du / dx)

·         dy / du = e u and du / dx = 2

f '(x) = (e u)(2) = 2 e u

·         Substitute u = 2x + 1 in f '(x) above

f '(x) = 2 e 2x + 1
Exercises Find the derivative of each function.

1 - f(x) = e x 2 x

2 - g(x) = 3 x - 3x 3

3 - h(x) = e x / (2x - 3)

4 - j(x) = e (x2 + 2)

solutions to the above exercises

1 - f '(x) = e x 2 x ( ln 2 + 1)

2 - g '(x) = 3 x ln 3 - 9x 2

3 - h '(x) = e x(2x -5) / (2x - 3) 2

4 - j '(x) = 2x e (x2 + 2)
   
Solve the following problem
Differentiate y=e^(ax+b) where a and b are constants
Let u = ax + b and y = e ^u
 Use the chain rule to find the derivative of function as follows 
 dy/dx = (dy / du) (du / dx) 
dy / du = e ^u and du / dx = a  ( Since the derivative of the constant b is 0)

Hence dy/dx = (e ^u)(a) = a e ^u
Substitute u = ax + b in dy/dx above 

dy/dx = a e ^ax + b


VOLUME OF A TRIANGULAR PRISM

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How Do You Find the Volume of a Triangular Prism?

Find the volume of the triangular prism below

SUMMARY

  1. B stands for the area of the base
  2. Since our bases are triangles, we can use (1/2)bh for B
  3. We can call the height of the triangle h1 and the height of the prism h2 to avoid confusion
  4. Volume is measured in units cubed, so we have 50 cm3
 

NOTES

  1.  
    1. The volume of a prism is V=Bh
    2. B is the area of the base of the prism
    3. h is the height of the prism
    4. The base of our prism is a triangle, so we can use (1/2)bh, the area of a triangle, for B
    5. We can call the height of the triangle h1 and the height of the prism h2 to avoid confusion
  2.  
    1. V is the volume, which is what we are trying to find
    2. b is the base of the triangle, which is 5 cm
    3. h1 is the height of the triangle, which is 2 cm
    4. h2 is the height of the prism, which is 10 cm
  3.  
    1. We can plug 5 in for 'b', 2 in for 'h1', and 10 in for 'h2' in our equation
  4.  
    1. We need to use the order of operations to simplify the right hand side
    2. First multiply 5•2 in the innermost parentheses
    3. Then multiply (1/2)•10 within the next set of parentheses
    4. This is the same as 10/2, or 5
    5. Then multiply 5•10 to get 50 cm3

VOLUME OF A RECTANGULAR PRISM


Volume of a rectangular prism or box examples: Let's do some example problems together in which we use the area times length formula for finding volume.