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Questions and Answers
What is the limit definition of the derivative for the function $f(x) = \sqrt{2x}$?
The limit definition is $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.
How can one show that the line $y = 2$ is a tangent line to the graph of $f(x) = \sqrt{2x}$?
To show the line $y=2$ is tangent, find the point where $f(c) = 2$ and check if the slope of the line equals the derivative at that point.
In the equation $4x^2 + y^2 = 4$, how do you determine the tangent lines passing through the point $(2, 0)$?
Implicit differentiation of $4x^2 + y^2 = 4$ gives the slope of the tangent line at points on the curve; then set up a point-slope form from $(2, 0)$.
What method can be used to find the derivatives of the functions $\cos(x^2) \sin(e^x)$ and $x^{1/x} \ln(x)$?
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What specific features should be indicated when sketching a curve, such as asymptotes and local maxima?
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Why is it important to show working when deriving answers in calculus?
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What is the significance of finding all tangent lines to a given curve?
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Explain how to find the local maxima of a function derived from implicit relations like $4x^2 + y^2 = 4$.
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Study Notes
Limit Definition of Derivative
- Calculate the derivative of a function using the limit definition.
- The derivative of a function f(x) is defined as the limit of the difference quotient as h approaches 0.
- f'(x) = lim(h->0) [f(x + h) - f(x)] / h.
Tangent Line
- Find the tangent line to a point on a graph using the derivative.
- If a line is tangent to the curve y = f(x) at the point (a, f(a)), it must have a slope equal to f'(a).
Derivatives Of Functions
- Determine the derivatives of functions using differentiation rules.
- Product Rule: The derivative of the product of two functions is the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.
- Chain Rule: The derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
Curve Sketching
- Sketch a curve by determining its asymptotes, local maxima and minima, and concavity.
- Asymptotes are lines that the curve approaches as x approaches infinity or negative infinity.
- Local maxima and minima are points where the curve changes from increasing to decreasing or decreasing to increasing.
- Concavity refers to whether the curve is concave up or concave down.
Curve Using Implicit Differentiation
- Find the equation of a tangent line to a curve given by an implicit equation.
- Implicit differentiation is used to find the derivative of a function implicitly defined by an equation.
- Solve for dy/dx to find the slope of the tangent line.
- Substitute the point of interest into the equation for dy/dx to find the slope of the tangent line at that point.
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Description
Test your understanding of derivatives using the limit definition, and find the tangent line to a point on a graph. This quiz covers differentiation rules including the product and chain rules, and it challenges you to apply these concepts in curve sketching.
