Lesson 11 The Limit Definition of the Derivative
11.1 Lecture Content
Video: Introduction to the Derivative
Video: Powers, Sums and Constant Multiples
11.2 Lecture Notes
11.2.1 Limit Definition of the Derivative
The derivative of a function \(f(x)\) at a point \(x = a\) is defined as:
\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]
Alternatively, using \(x\):
\[ f'(x) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]
This represents the instantaneous rate of change or the slope of the tangent line to the curve at \(x = a\).
11.2.2 Derivative Rules for Common Functions
11.2.3 Sum, Difference, and Constant Multiple Rules
Sum Rule:
\[ \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) \]Difference Rule:
\[ \frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x) \]Constant Multiple Rule:
\[ \frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x) \]
These rules allow for simple combination and manipulation of derivative calculations.
11.2.4 Tangent Line to a Function at a Point
To find the equation of the tangent line to \(f(x)\) at \(x = a\):
- Find \(f(a)\) – the point of tangency: \((a, f(a))\).
- Compute \(f'(a)\) – the slope of the tangent line.
- Use point-slope form:
\[ y - f(a) = f'(a)(x - a) \]
This gives the linear approximation of the function near \(x = a\).
Example:
Find the tangent line to \(f(x) = x^2\) at \(x = 3\):
- \(f(3) = 9\)
- \(f'(x) = 2x \Rightarrow f'(3) = 6\)
- Equation: \(y - 9 = 6(x - 3) \Rightarrow y = 6x - 9\)
11.3 Examples
11.3.1 Average ROC
- The following chart shows “living wage” jobs in Rochester per 1000 working-age adults over a 5-year period.
| Year | 1997 | 1998 | 1999 | 2000 | 2001 |
|---|---|---|---|---|---|
| Jobs | 660 | 735 | 795 | 840 | 870 |
What is the average rate of change in the number of living wage jobs from 1997 to 1999?
What is the average rate of change in the number of living wage jobs from 1999 to 2001?
Find the slope of the line through the points \((-3,0)\) and \((1,3)\).
An object is moving along a vertical line. Its vertical position is given by: \[ L(t) = t^3 -4t^2 - 4t + 10 \] where distance is measured in meters and time in seconds. Find the approximate average velocity (accurate to at least 3 decimal places) in each time interval:
a.Between \(t_1 = 2 \ \text{s}\) and \(t_2 = 7 \ \text{s}\)
Between \(t_1 = 1 \ \text{s}\) and \(t_2 = 7 \ \text{s}\)
Between \(t_1 = 4 \ \text{s}\) and \(t_2 = 5 \ \text{s}\)
Let \(f(x) = 3 \ln(x)\). Using the formula: \[ m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \] with \(x_1 = 7\), find the slope of the secant line for each \(x_2\) value listed below:
| \(x_2\) | \(m\) (slope) |
|---|---|
| 12 | \(\_\_\_\_\_\_\) |
| 9 | \(\_\_\_\_\_\_\) |
| 8 | \(\_\_\_\_\_\_\) |
| 7.1 | \(\_\_\_\_\_\_\) |
| 7.01 | \(\_\_\_\_\_\_\) |
11.3.2 Average ROC
Suppose that
\(f(x) = -2x^2 - 2x -4\). Find the derivative of \(f\).Let \(f(x) = 12x^2 - 7x + 3\). Simplify the difference quotient: \[ \frac{f(3 + h) - f(3)}{h} \]
If \(f(x) = -3x^2 + 7x + 7\), find the instantaneous rate of change of \(f(x)\) at \(x = -3\).
If \(f(x) = \frac{4}{x^2}\), find \(f'(3)\).
11.3.3 Derivatives using Limits
Given \(f(x)=5 x^{2}+ 12x+7\), find \(f'(x)\) using the limit definition of the derivative.
Let \[ f(x)=3x^{3} + 7x - 4. \] Use the limit definition of the derivative to calculate the derivative and second derivative of \(f\).
For what values of \(A\) and \(B\) is the function \(f(x)\) differentiable at \(x=-2\)?
\[ f(x)= \begin{cases} Ax+B, & x< 2,\\[6pt] x^{2}+x, & x\ge 2. \end{cases} \]
- Use the limit definition to find the derivative of \[ f(x)=\sqrt{x+5}. \]
11.3.4 Basic Derivative Rules
Find the derivative of the function \(f(x) = 7x^{3} - 10x^{7}\).
If \(f(x) = \frac{4}{x^2\), find the value of the derivative at \(x=1\):
Find the derivative of the function: \(f(x) = 2\sqrt{x} - \frac{9}{x^9}\).
4.Given \(f(x) = 4x\sqrt{x} + \frac{5}{5x^{2}\sqrt{x}}\), find \(f'(x)\) and then compute \(f'(1) =\)
Let \(h(t) = 5t^{3.2} - 8t^{-3.2}\). Find \(h'(t)\), \(h'(4)\), \(h''(t)\), and \(h''(4)\)
If \[ g(t) = -2t^{4} + 5t^{2} + 5, \] find the following values: \[ \begin{aligned} g(0) &= \underline{\qquad\qquad} \\ g'(0) &= \underline{\qquad\qquad} \\ g''(0) &= \underline{\qquad\qquad} \\ g'''(0) &= \underline{\qquad\qquad} \\ g^{(4)}(0) &= \underline{\qquad\qquad} \\ g^{(5)}(0) &= \underline{\qquad\qquad} \end{aligned} \]
11.4 Practice Problems
Practice the techniques discussed in class and in the online videos by solving the following examples.
Use the limit definition of a derivative to find the derivative of \(f(x) = x^{2} - 3\).
Use the limit definition of a derivative to find the derivative of \(f(x) = \frac{2}{x^{2}}\).
Differentiate the following functions:
\(f(x) = x^{5} + 2x - 2\).
\(f(x) = |x| + \frac{1}{x} + \ln(x)\).
\(f(x) = \log_{3} x\).
\(g(x) = 5^{x} + e^{x}\).
\(y = 4x^{-1} - 2|x| - 10 + \sin(x)\).
\(f(x) = 3x^{3} - 2x^{2} + \sqrt{x}\).
\(f(x) = 5 \sin(x) + \ln(x)\).
\(f(x) = \cos(x) - 3 \sin(x) + e^{x}\).
Find the equation of the tangent line to the equation below at the indicated point:
\(f(x) = x + \frac{1}{x}\) at \(x=2\).
\(f(x) = \frac{1}{x^{2}}\) at \(x=1\).
\(f(x) = \sqrt{x}\) at \(x=4\).
(Applied) Company C’s profits are given by \(P(0) = \$1\) million and \(P'(0) = \$0.5\) million/month. Company D’s profits are given by \(P(0) = \$0.5\) million and \(P'(0) = \$1\) million/month. In which company would you rather invest? Why?
(Applied) The cost of producing \(x\) teddy bears per day at the Cuddly Companion Co. is calculated by their marketing staff to be given by the formula \[ C(x) = 100 + 40x - 0.001 x^{2}. \]
Find the marginal cost function and use it to estimate how fast the cost is going up at a production level of 100 teddy bears. Compare this with the exact cost of producing the 101st teddy bear.
Find the average cost function \(\overline{C}\), and evaluate \(\overline{C}(100)\). What does the answer tell you?
11.5 Self Assessment
Time yourself and try to solve the following questions within twenty minutes.
Use the limit definition of a derivative to find the derivative of \(f(x) = x - 2x^{3}\).
Differentiate the following functions:
\(f(x) = 2x^{4} + 3x^{3} - 1\).
\(y = x^{2} + 3|x| - \cos(x)\).
\(f(x) = x^{2} - 3\sqrt{x} + 5\).
\(f(x) = 5 \sin(x) + \ln(x)\).
\(f(x) = \cos(x) - 3 \sin(x) + e^{x}\).
Find the equation of the tangent line to \(f(x) = x^{2} + \cos(x)\) at \(x=0\).
Daily oil production by Pemex, Mexico’s national oil company, can be approximated by \[ P(t) = -0.022 t^{2} + 0.2 t + 2.9 \text{ million barrels } \quad (1 \leq t \leq 9), \] where \(t\) is time in years since the start of 2000. Find the derivative function \(dP/dt\). At what rate was oil production changing at the start of 2004 (\(t=4\))?
A car wash firm calculates that its daily profit (in dollars) depends on the number \(n\) of workers it employs according to the formula \[ P = 400 n - 0.5 n^{2}. \] Calculate the marginal product at an employment level of 50 workers, and interpret the result.
11.6 Lesson Checklist
This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.
\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]
| Skill | D | CON | COM |
|---|---|---|---|
| Use the limit definition to find the derivative of a function. | |||
| Find the derivatives of polynomial and absolute value functions. | |||
| Find the derivatives of exponential and logarithmic functions. | |||
| Find the derivative of trigonometric functions. | |||
| Determine the equation of a tangent line at a given point. | |||
| Solve applied problems involving derivatives. |