Determine whether the following functions are differentiable over their entire domain:
Consider the graph of the function \\ f \left( x \right) = 2 x + 2 shown:
Are there any points within its domain where the function is discontinuous?
Is the function continuous over its domain?
Is the function smooth?
State the domain over which the function is differentiable.
For each of the following graphs of the functions:
Are there any points within its domain where the function is discontinuous?
Is the function continuous over its domain?
State the domain over which the function is differentiable.
For each of the following functions:
Sketch the graph of the function.
Using interval notation, state the domain over which the function is differentiable.
f \left( x \right) = \sqrt{36 - x^{2}}
f \left( x \right) = \left| 2 x + 4\right| + 3
The water level in a dam is measured to be 190 \text{ m}. After two weeks, the water level is measured again and found to be 140 \text{ m}.
Is there a time during the two weeks that the water level of the dam is at 160 \text{ m}?
To answer part (a) what property do we assume about the water level over the two weeks?
Consider a function f \left( x \right) with domain \left[0, 2\right], where f \left( 0 \right) = 190 and f \left( 2 \right) = 140. If the function is known to be discontinuous, does the function take a value of 160 at some point in its domain?
Consider the function f \left( x \right) = 4 x.
Does the function have a constant gradient?
\left(1, 4\right) and \left(7, 28\right) are two points that satisfy the function. Determine the gradient of the function using these two points.
Determine the gradient of the function from first principles.
For each of the following functions, find:
f \left( x + h \right)
f \left( x + h \right) - f \left( x \right)
\dfrac{f \left( x + h \right) - f \left( x \right)}{h}
f \left( x \right) = 3 x + 2
f \left( x \right) = 8 - x
f \left( x \right) = - 5 x + 4
f \left( x \right) = x^{2}
f \left( x \right) = 5 - x^{2}
f \left( x \right) = 3 x^{2} - 4 x + 2
f \left( x \right) = x^{2} + 3 x + 6
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}.
Hence, find f' \left( x \right) from first principles by evaluating \lim_{h \to 0}\left(\dfrac{f \left( x + h \right) - f \left( x \right)}{h}\right).
Find the derivative of the following functions from first principles:
f \left( x \right) = 6
f \left( x \right) = - 6
f \left( x \right) = 5 x
f \left( x \right) = x^{3}
f \left( x \right) = 2 x + 3
f \left( x \right) = x^{2} - 3 x
f \left( x \right) = 4 x^{2} - 3 x - 5
Consider the function f \left( x \right) = 7 - x^{2}.
Find an expression for the derivative 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 an expression for the derivative 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) = \left(x + 6\right) \left(x + 2\right).
Find an expression for the derivative f' \left( x \right) using first principles.
Hence, calculate the gradient at the point where x = 3.
Consider the function f \left( x \right) = x^{3}.
Find an expression for the derivative f' \left( x \right) using first principles.
Hence, calculate the gradient of f \left( x \right) at the point where x = 2.
Use first principles to find the gradient of f \left( x \right) = 3 x^{2} at the point \left(4, 48\right).
Use first principles to find the gradient of f \left( x \right) = 2 x \left(x - 6\right) at the point \left(2, - 16 \right).
Use first principles to find the gradient of f \left( x \right) = 3 x^{3} at the point \left( - 4 , - 192 \right).