Examples of slope in the following topics:
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- The use of differentiation makes it possible to solve the tangent line problem by finding the slope $f'(a)$.
- The slope of the secant line passing through $p$ and $q$ is equal to the difference quotient
- If $k$ is known, the equation of the tangent line can be found in the point-slope form:
- It barely touches the curve and shows the rate of change slope at the point.
- Define a derivative as the slope of the tangent line to a point on a curve
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- Thus, to solve the tangent line problem, we need to find the slope of a line that is "touching" a given curve at a given point, or, in modern language, that has the same slope.
- But what exactly do we mean by "slope" for a curve?
- In this case, $y = f(x) = m x + b$, for real numbers m and b, and the slope m is given by:
- This gives an exact value for the slope of a straight line.
- If $x$ and $y$ are real numbers, and if the graph of $y$ is plotted against $x$, the derivative measures the slope of this graph at each point.
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- Direction fields, also known as slope fields, are graphical representations of the solution to a first order differential equation.
- The slope field is traditionally defined for differential equations of the following form:
- An isocline (a series of lines with the same slope) is often used to supplement the slope field.
- Then, from the differential equation, the slope to the curve at $A_0$ can be computed, and thus, the tangent line.
- Along this small step, the slope does not change too much $A_1$ will be close to the curve.
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- The shape of a graph may be found by taking derivatives to tell you the slope and concavity.
- This gives an exact value for the slope of a straight line.
- At each point, the derivative of is the slope of a line that is tangent to the curve.
- The line is always tangent to the blue curve; its slope is the derivative.
- Sketch the shape of a graph by using differentiation to find the slope and concavity
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- Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.
- Its slope is the velocity at that point.
- Recognize that the slope of a tangent line to a curve gives the instantaneous velocity at that point in time
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- Partial differentiation is the act of choosing one of these lines and finding its slope.
- To find the slope of the line tangent to the function at $P(1, 1, 3)$ that is parallel to the $xz$-plane, the $y$ variable is treated as constant.
- By finding the derivative of the equation while assuming that $y$ is a constant, the slope of $f$ at the point $(x, y, z)$ is found to be:
- So at $(1, 1, 3)$, by substitution, the slope is $3$.
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- You can use implicit differentiation to find the slope of a line tangent to the circle at a point $(x,y)$.
- Since the slope of a tangent is the derivative at that point, we find the derivative implicitly:
- and you can now find the slope at any point $(x,y)$.
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- Visually, derivative of a function $f$ at $x=a$ represents the slope of the curve at the point $x=a$.
- The slope of tangent line shown represents the value of the derivative of the curved function at the point $x$.
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- This is read as "$f$ double prime of $x$," or "the second derivative of $f(x)$. " Because the derivative of a function is defined as a function representing the slope of function, the double derivative is the function representing the slope of the first derivative function.
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- The slope of the graph at any point is the height of the function at that point.
- Since the slope of the red tangent line (the derivative) at $P$ is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function, $h$ must be equal to the ratio of $h$ to $b$.