Problem 29
Question
Use the limit definition to find the derivative of the function. $$ g(s)=\frac{1}{3} s+2 $$
Step-by-Step Solution
Verified Answer
The derivative of the function is \(\frac{1}{3}\).
1Step 1: Define the function
Firstly, identify the function which is \(g(s)=\frac{1}{3} s+2\).
2Step 2: Apply the limit definition of derivative
Use the definition of the derivative which is the limit as h approaches 0 of (f(x+h) - f(x))/h. So in this case, (g(s+h) - g(s))/h is used.
3Step 3: Calculate g(s + h)
Calculate g(s + h) by replacing every s in the function with (s + h) which gives \(\frac{1}{3} (s + h) + 2\).
4Step 4: Substitute into the derivative definition
Substitute g(s + h) and g(s) into the derivative definition to get \([ \frac{1}{3} (s + h) + 2 - ( \frac{1}{3} s + 2 ) ] / h\). Simplify this to get \(\frac{1}{3}\).
5Step 5: Calculate the limit
As this doesn't depend on h anymore, the limit as h approaches 0 is simply \(\frac{1}{3}\).
Other exercises in this chapter
Problem 29
Find the marginal profit for producing \(x\) units. (The profit is measured in dollars.) $$ P=-0.00025 x^{2}+12.2 x-25,000 $$
View solution Problem 29
Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, id
View solution Problem 29
Find the limit. $$ \lim _{x \rightarrow 3} \sqrt{x+6} $$
View solution Problem 30
Use the General Power Rule to find the derivative of the function. $$ f(t)=(9 t+2)^{2 / 3} $$
View solution