Consider the function f \left( x \right) = x^{3}.
Complete the limiting chord process for f \left( x \right) = x^{3} at the point x = 3 by filling in the table of values. Round your answers to four decimal places.
| a | b | h=b-a | \dfrac{f\left(b\right)-f\left(a\right)}{b-a} |
|---|---|---|---|
| 3 | 4 | 1 | |
| 3 | 3.5 | ||
| 3 | 3.1 | ||
| 3 | 3.05 | ||
| 3 | 3.01 | ||
| 3 | 3.001 | ||
| 3 | 3.0001 |
Use the limiting chord process to calculate the instantaneous rate of change for the remaining values of x in the table:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Instantaneous rate of change} f(x) \text{ at } x | 3 | 12 | 48 |
Find the instantaneous rate of change of f \left( x \right) at any point x.
Consider the function f \left( x \right) = x^{2}.
Complete the limiting chord process for f \left( x \right) = x^{2} at the point x = 3 by filling in the table of values:
| a | b | h=b-a | \dfrac{f\left(b\right)-f\left(a\right)}{b-a} |
|---|---|---|---|
| 3 | 4 | 1 | |
| 3 | 3.5 | ||
| 3 | 3.1 | ||
| 3 | 3.05 | ||
| 3 | 3.01 | ||
| 3 | 3.001 | ||
| 3 | 3.0001 |
Use the limiting chord process to calculate the instantaneous rate of change for the remaining values of x in the table:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Instantaneous rate of change} f(x) \text{ at } x | 2 | 4 | 8 |
Find the instantaneous rate of change of f \left( x \right) at any point x.
Consider the function f \left( x \right) = 2 x^{2}.
Complete the limiting chord process for f \left( x \right) = 2 x^{2} at the point x = 3 by filling in the table of values. Round your answers to four decimal places.
| a | b | h=b-a | \dfrac{f\left(b\right)-f\left(a\right)}{b-a} |
|---|---|---|---|
| 3 | 4 | 1 | |
| 3 | 3.5 | ||
| 3 | 3.1 | ||
| 3 | 3.05 | ||
| 3 | 3.01 | ||
| 3 | 3.001 | ||
| 3 | 3.0001 |
Use the limiting chord process to calculate the instantaneous rate of change for the remaining values of x in the table:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Instantaneous} \\\text{rate of change}\\ \text{ of }f(x) \text{ at } x | 4 | 8 | 16 |
Hence, find the instantaneous rate of change of f \left( x \right) for any given value of x.
For each of the following functions:
Find f \left( x + h \right).
Find f \left( x + h \right) - f \left( x \right).
Find \dfrac{f \left( x + h \right) - f \left( x \right)}{h}.
For each of the following functions:
Find f \left( x + h \right) in expanded form.
Find f \left( x + h \right) - f \left( x \right).
Find \dfrac{f \left( x + h \right) - f \left( x \right)}{h}.
Hence, find f' \left( x \right) by evaluating \lim_{h \to 0}\left(\dfrac{f \left( x + h \right) - f \left( x \right)}{h}\right).
Find the derivative of each of the following functions from first principles:
Consider the function f \left( x \right) = 4 x.
Is the gradient of the function constant or variable?
Find the gradient of the function using the points \left(1, 4\right) and \left(7, 28\right) that lie on the function.
Find the gradient of the function from first principles.
Consider the function f \left( x \right) = \left(x + 6\right) \left(x + 2\right).
Find the gradient function of f \left( x \right) = \left(x + 6\right) \left(x + 2\right) from first principles.
Find the gradient at the point where x = 3.
Consider the function y = \sqrt{x}.
Find the gradient function of f \left( x \right) = \sqrt{x} from first principles.
Find the gradient at the point where x = 25.
Consider the function f \left( x \right) = 7 - x^{2}.
Find the gradient function f' \left( x \right) using first principles.
Hence, calculate the gradient of f \left( x \right) at the point where x = -1.
Consider the function f \left( x \right) = x^{2} + 3 x + 4.
Find the gradient function f' \left( x \right) using first principles.
Hence, calculate the gradient of f \left( x \right) at the point where x = 4.
Consider the function f \left( x \right) = x^{3}.
Find the gradient function f' \left( x \right) using first principles.
Hence, calculate the gradient of f \left( x \right) at the point where x = 2.
Consider the function f \left( x \right) = x^{4}.
Find the gradient function f' \left( x \right) using first principles.
Hence, calculate the gradient of f \left( x \right) at the point where x = - 5.
Find the gradient of f \left( x \right) = 3 x^{2} at the point \left(4, 48\right) from first principles.
Find the gradient of f \left( x \right) = 3 x^{3} at the point \left( - 4 , - 192 \right) from first principles.
Find the gradient of f \left( x \right) = 2 x \left(x - 6\right) at the point \left(2, - 16 \right) from first principles.
Find the gradient of f \left( x \right) = 3 x \left(x - 2\right)^{2} at the point \left(4, 48\right) from first principles.
Consider the function y = \dfrac{1}{x}.
Find the gradient function of y = \dfrac{1}{x} from first principles.
Find the x-values of the two points on the function at which the gradient is - \dfrac{1}{16}.
Consider the function f \left( x \right) = \dfrac{1}{2 x - 4}.
Find f' \left( x \right) from first principles.
Hence, find f' \left( 4 \right).