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Derivatives

In finance, a derivatives contract is a contract that describes how to buy or sell an asset at a specified price on a specified future date. Derivatives are used to hedge against price changes, interest rate risk or market risk.

Slopes and Rates of Change

462 Practice Problems
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01:07
21st Century Astronomy

When viewed by radio telescopes, Jupiter is the second-brightest object in the sky. What is the source of its radiation?

Worlds of Gas and Liquid-The Giant Planets
05:25
General Chemistry: Principles and Modern Applications

[A] $_{t}$ as a function of time for the reaction $A \longrightarrow$ products is plotted in the following graph. Use data from this graph to determine (a) the order of the reaction; (b) the rate constant, $k ;$ (c) the rate of the reaction at $t=3.5$ min, using the results of parts (a) and (b); (d) the rate of the reaction at $t=5.0 \mathrm{min}$, from the slope of the tangent line; (e) the initial rate of the reaction.

Chemical Kinetics
David Collins
0:00
Calculus for Scientists and Engineers: Early Transcendental

Average and marginal profit Let $C(x)$ represent the cost of producing $x$ items and $p(x)$ be the sale price per item if $x$ items are sold. The profit $P(x)$ of selling $x$ items is $P(x)=x p(x)-C(x)$ (revenue minus costs). The average profit per item when $x$ items are sold is $P(x) / x$ and the marginal profit is dP/dx. The marginal profit
approximates the profit obtained by selling one more item given that $x$
items have already been sold. Consider the following cost functions $C$ and price functions $p$
a. Find the profit fiunction $P$.
b. Find the average profit function and marginal profit function.
c. Find the average profit and marginal profit if $x=$ a units have been sold.
d. Interpret the meaning of the values obtained in part (c).
$$\begin{aligned}
&C(x)=-0.04 x^{2}+100 x+800, p(x)=200-0.1 x\\
&a=1000
\end{aligned}$$

Derivatives
Derivatives as Rates of Change

Tangent Lines

398 Practice Problems
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02:30
Calculus: Early Transcendental Functions

In exercises identify all points at which the curve has
(a) a horizontal tangent and (b) a vertical tangent.
$$\left\{\begin{array}{l}
x=t^{2}-1 \\
y=t^{4}-4 t
\end{array}\right.$$

Parametric Equations and Polar Coordinates
Calculus and Parametric Equations
John Irizar
03:17
Calculus: Early Transcendental Functions

In exercises find the slopes of the tangent lines to the given curves at the indicated points.
$$\left\{\begin{array}{l}
x=t^{2}-2 \\
y=t^{3}-t
\end{array}\right.$$

Parametric Equations and Polar Coordinates
Calculus and Parametric Equations
John Irizar
01:13
Calculus: Early Transcendental Functions

Find all values of $t$ such that $\mathrm{r}^{\prime}(t)$ is parallel to the $x y$ -plane.
$$\mathbf{r}(t)=\langle\cos t, \sin t, \sin 2 t\rangle$$

Vector-Valued Functions
The calculus of Vector-Valued Functions
Ernest Castorena

Derivative as a Function

974 Practice Problems
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01:42
Calculus for Scientists and Engineers: Early Transcendental

Consider the general parabola described by the function $f(x)=a x^{2}+b x+c .$ For what values of $a, b,$ and $c$ is $f$ concave up? For what values of $a, b,$ and $c$ is $f$ concave down?

Applications of the Derivative
What Derivatives Tell Us
Rakesh Kumar Sharma
03:37
Calculus for Scientists and Engineers: Early Transcendental

The functions $f(x)=a x^{2},$ where $a > 0$ are concave up for all $x$. Graph these functions for $a=1,5,$ and $10,$ and discuss how the concavity varies with $a .$ How does $a$ change the appearance of the graph?

Applications of the Derivative
What Derivatives Tell Us
Rakesh Kumar Sharma
02:58
Calculus for Scientists and Engineers: Early Transcendental

Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine whether they correspond to local maxima, local minima or neither.
$$f(x)=2 x^{2} \ln x-11 x^{2}$$

Applications of the Derivative
What Derivatives Tell Us
Rakesh Kumar Sharma

Concavity

153 Practice Problems
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03:03
Calculus: Early Transcendental Functions

Suppose that a company that spends $\$ x$ thousand on advertising sells $\$ s(x)$ of merchandise, where $s(x)=-3 x^{3}+270 x^{2}-3600 x+18,000 .$ Find the value of
$x$ that maximizes the rate of change of sales. (Hint: Read the question carefully!) Find the inflection point and explain why in advertising terms this is the "point of diminishing returns."

Applications of Differentiation
Concavity and The Second Derivative Test
Jonathon Brumley
03:20
Calculus: Early Transcendental Functions

The "family of functions" contains a parameter $c .$ The value of $c$ affects the properties of the functions. Determine what differences, if any, there are for $c$ being zero, positive or negative. Then determine what the graph would look like for very large positive $c$ 's and for very large negative $c$ 's.
$$f(x)=x^{4}+c x^{2}$$

Applications of Differentiation
Overview of Curve Sketching
Aaron Wan
05:47
Calculus: Early Transcendental Functions

Determine all significant features by hand and sketch a graph.
$$f(x)=\left(x^{2}+1\right)^{2 / 3}$$

Applications of Differentiation
Concavity and The Second Derivative Test
Jonathon Brumley

Extrema

289 Practice Problems
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08:04
Calculus: Early Transcendental Functions

Locate all critical points and classify them using Theorem 7.2.
$f(x, y)=e^{-x^{2}}\left(y^{2}+1\right)$

Functions of Several Variables and Partial Differentiation
Extrema of Functions of Several Variables
Jeffrey Utley
06:44
Calculus: Early Transcendental Functions

Suppose a painting hangs on a wall as in the figure. The frame extends from 6 feet to 8 feet above the floor. A person whose eyes are 5 feet above the ground stands $x$ feet from the wall and views the painting, with a viewing angle $A$ formed by the ray from the person's eye to the top of the frame and the ray from the person's eye to the bottom of the bottom of the frame. Find the value of $x$ that maximizes the viewing angle $A$
(FIGURE CANNOT COPY)

Applications of Differentiation
Maximum and Minimum Values
Sriram Soundarrajan
02:22
Calculus: Early Transcendental Functions

For the family of functions $f(x)=x^{4}+c x^{2}+1,$ find all local extrema (your answer will depend on the value of the constant $c)$

Applications of Differentiation
Maximum and Minimum Values
Rebecca Polasek

Local vs Global (Absolute) Extrema: 1st Derivative Test

134 Practice Problems
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01:18
Calculus: Early Transcendental Functions

Determine all significant features (approximately if necessary) and sketch a graph.
$$f(x)=\sin x-\frac{1}{2} \sin 2 x$$

Applications of Differentiation
Overview of Curve Sketching
Mutahar Mehkri
02:04
Calculus: Early Transcendental Functions

Find (by hand) all critical numbers and use the First Derivative Test to classify each as the location of a local maximum, local minimum or neither.
$$y=x^{2} e^{-x}$$

Applications of Differentiation
Increasing and Decreasing Functions
Stephen Hobbs
02:14
Calculus: Early Transcendental Functions

Determine all significant features (approximately if necessary) and sketch a graph.
$$f(x)=\left(x^{3}-3 x^{2}+2 x\right)^{2 / 3}$$

Applications of Differentiation
Overview of Curve Sketching
Mutahar Mehkri

Local vs Global (Absolute) Extrema: 2nd Derivative Test

81 Practice Problems
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02:59
Calculus: Early Transcendental Functions

Prove Theorem 5.2 (the Second Derivative Test). (Hint: Think about what the definition of $f^{\prime \prime}(c)$ says when $f^{\prime \prime}(c) > 0$ or $\left.f^{\prime \prime}(c) < 0 .\right)$

Applications of Differentiation
Concavity and The Second Derivative Test
Jonathon Brumley
00:43
Calculus: Early Transcendental Functions

Estimate the intervals where the function is concave up and concave down. (Hint: Estimate where the slope is increasing and decreasing.)
(GRAPH CAN'T COPY)

Applications of Differentiation
Concavity and The Second Derivative Test
Jonathon Brumley
06:16
Calculus: Early Transcendental Functions

Determine all significant features (approximately if necessary) and sketch a graph.
$$f(x)=2 x^{4}-11 x^{3}+17 x^{2}$$

Applications of Differentiation
Concavity and The Second Derivative Test
Jacquelyn Trost

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