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 derivatives basics of graphing

external links you may find helpful related topics on other pages increasing/decreasing intervals critical points first derivative test graphing youtube playlist

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

On this page we discuss using the second derivative to help determine what the graph of a function looks like. The techniques discussed are concavity, inflection points and the second derivative test.

Concavity

Concavity relates to how the graph is curving, either upward or downward. If a graph is curving upward, then it looks like a cup and it could hold water. The graph of $$y=x^2$$ is concave upward.

If the graph is curving downward, then it looks like an arched roof and could keep you dry you in the rain. The graph of $$y=-x^2$$ is concave downward.

This plot shows several examples of concavity to help you get a feel for what they look like.

The concept of concavity parallels increasing and decreasing intervals. We learned that increasing and decreasing sections of a function change only at critical points. Similarly, concavity is related to the second derivative and changes only at inflection points.

Here is a quick video clip explaining this idea again.

### PatrickJMT - Concavity [10min-22secs]

video by PatrickJMT

Inflection Points

We use inflection points to help us determine the where concavity changes. Basically, concavity will change only at inflection points. To find inflection points, we use a similar procedure as we did for critical points, except we us the second derivative. So we start by taking the derivative twice, set the result to zero and solve for the x-values. We also look at values where the second derivative is not defined but the points are in the domain of the original function.

The inflection points allow us to determine concavity. We can use the following table format to organize the information. We assume here that we have a function $$g(x)$$ with break points at $$x=c_1$$, $$x=c_2$$ and $$x=c_3$$ and the function is defined for $$x < c_1$$ and for $$x > c_3$$.

Table Format For Inflection Points

Interval

$$-\infty < x < c_1$$

$$c_1 < x < c_2$$

$$c_2 < x < c_3$$

$$c_3 < x < \infty$$

Test x-value

Sign of $$g''(x)$$

Conclusion

Possible conclusions include concave upward or concave downward.
Note: The break points include points of inflection and discontinuities. Basically, the entire domain needs to be covered by the intervals in the first row.

The test values can be any point in the open interval in each column.

Before we go on, try these practice problems. Unless otherwise instructed, find the points of inflection and determine concavity.

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new.

Second Derivative - Practice Problems Conversion

[A01-1349] - [A02-1350] - [A03-1351] - [A04-1348] - [A05-1354] - [B01-1352] - [B02-1353]

Please update your notes to this new numbering system. The display of this conversion information is temporary.

GOT IT. THANKS!

concavity, inflection
$$f(x)=2x^3+6x^2-5x+1$$

Problem Statement

concavity, inflection
$$f(x)=2x^3+6x^2-5x+1$$

Solution

### 1349 solution video

video by Krista King Math

concavity, inflection
$$\displaystyle{f(x)=\frac{x^2+1}{x^2}}$$

Problem Statement

concavity, inflection
$$\displaystyle{f(x)=\frac{x^2+1}{x^2}}$$

Solution

### 1350 solution video

video by Krista King Math

concavity, inflection
$$f(x)=2+3x^2-x^3$$

Problem Statement

concavity, inflection
$$f(x)=2+3x^2-x^3$$

Solution

### 1351 solution video

video by PatrickJMT

concavity, inflection
$$h(x)=(x^2-1)^3$$

Problem Statement

concavity, inflection
$$h(x)=(x^2-1)^3$$

Solution

### 1352 solution video

video by PatrickJMT

Determine the open intervals where the function $$f(x)=x^2 e^{4x}$$ is concave up or concave down and determine any points of inflection.

Problem Statement

Determine the open intervals where the function $$f(x)=x^2 e^{4x}$$ is concave up or concave down and determine any points of inflection.

Solution

### 2290 solution video

video by MIP4U

Second Derivative Test

The second derivative test is interesting in that it tests for the same information as the first derivative test. On the good side it is easier to use. On the bad side, it doesn't always work.

The idea is that you find the second derivative and then plug the critical points in the second derivative. If the result is less than zero, then you have a relative maximum, greater than zero, a relative minimum. If you get zero, the test is inconclusive and you drop back and use the first derivative test. To organize your information, you can use a table in this format.

Table Format For The Second Derivative Test

Interval/Point

$$x = c_1$$

$$x = c_2$$

$$x = c_3$$

Sign of $$g''(x)$$

Conclusion

The possible conclusions in the last row above maximum, minimum or inconclusive. In the last case, we need to use the first derivative test.

Before working practice problems, take a few minutes to watch this quick video clip explaining the second derivative test in more detail.

### PatrickJMT - Second Derivative Test [1min-30secs]

video by PatrickJMT

For these problems, use the second derivative test to determine maximums and minimums, unless otherwise instructed.

maximums/minimums
$$f(x)=x^3-3x+1$$

Problem Statement

maximums/minimums
$$f(x)=x^3-3x+1$$

Solution

### 1348 solution video

video by Krista King Math

Suppose $$f(x)$$ has a critical point at $$x=4$$ and $$f''(4) = -3$$. What can be said about $$f(x)$$ at $$x=4$$?

Problem Statement

Suppose $$f(x)$$ has a critical point at $$x=4$$ and $$f''(4) = -3$$. What can be said about $$f(x)$$ at $$x=4$$?

Solution

### 1354 solution video

video by PatrickJMT

maximums/minimums
$$\displaystyle{f(x)=\frac{x}{x^2+4}}$$

Problem Statement

maximums/minimums
$$\displaystyle{f(x)=\frac{x}{x^2+4}}$$

Solution

### 1353 solution video

video by PatrickJMT