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# 2.4. Univariate Calculus: Slope, curvature and optimization

### Learning Objectives

You should be able to

• Determine on what regions a function is increasing or decreasing by considering the sign of the derivative
• Compute the equations of the tangent and normal lines to a curve of the form $$y=f(x)$$
• Determine on what regions a function is concave or convex
• Find and classify stationary points
• Find (global) maximum and minimum of a function by considering stationary points, boundaries and singularities

### Get Started: What to do next

If you can pass the diagnostic test, then we believe you are ready to move on from this topic.
Our recommendation is to:
1. Test your ability by trying the diagnostic quiz. You can reattempt it as many times as you want and can leave it part-way through.
2. Identify which topics need more work. Make a note of any areas that you are unable to complete in the diagnostic quiz, or areas that you don't feel comfortable with.
3. Watch the tutorials for these topics, you can find these below. Then try some practice questions from the mini quiz for that specific topic. If you have questions, please ask on the forum.
4. Re-try the diagnostic quiz. And repeat the above steps as necessary until you can pass the diagnostic test. A pass grade is 80%.

Optionally, also see if you can apply these skills to an economic application or look at the additional resources and advanced quizzes at the bottom of the page,

### Tutorials: How to guides

#### 2.4.1. Monotonicity (increasing and decreasing)

$$f'>0 \implies f$$ (strictly) increasing, $$f'\geq0 \implies f$$ (weakly) increasing,
$$f'<0 \implies f$$ (strictly) decreasing, $$f'\leq0 \implies f$$ (weakly) decreasing,

#### 2.4.2. Tangents and normals

Equation of tangent to $$y=f(x)$$ when $$x_0$$ is $$y-f(x_0) = f'(x_0)(x-x_0)$$
Equation of normal to $$y=f(x)$$ when $$x_0$$ is $$y-f(x_0) = \frac{-1}{f'(x_0)}(x-x_0)$$

#### 2.4.3. Curvature (convexity and concavity)

$$f''>0 \implies$$ convex, $$f''<0 \implies$$ concave
Point of inflexion when curvature changes

#### 2.4.4. Finding stationary points

First order condition: $$f'(x_0) = 0$$

#### 2.4.5. Classifying stationary points

At stationary point, $$f''>0 \implies$$ local minimum
At stationary point, $$f''<0 \implies$$ local maximum

#### 2.4.6. Optimisation

The difference between stationary points, turning points and critical points.
The process of finding the (global) maximum and minimum of a function of
one variable by considering stationary points, boundaries and singularities.

### Economic Application: Economies of scale; profit maximization; slope and curvature of production function

This economic application explains the importance of using derivative for studying functions in the context of three important economic applications. The video introduces the concept of average cost functions and economies (and diseconomies) of scale and explains how these can be characterized by using derivatives. The quiz gives further examples of economic context in which optimization and characterising the slope and curvature of functions is important in economics.

#### Economic Application Exercise

The following quiz allows you to test your understanding of the theory underlying the discussion of economies and diseconomies of scale. It further provides an example of a monopolist's profit maximization problem, and introduces the concepts of positive and diminishing marginal products from mathematical perspective.

### Further practice & resources

You can find practice quizzes on each topic in the links above the tutorial videos, or you can take the diagnostic quiz as many times as you like to If you want to test yourself further, then try the advanced quiz linked below.

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