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Questions and Answers
What does the difference quotient represent?
What does the difference quotient represent?
What happens to the slope of a secant line as the distance between the two points approaches zero?
What happens to the slope of a secant line as the distance between the two points approaches zero?
Which expression correctly represents the limit definition of a derivative?
Which expression correctly represents the limit definition of a derivative?
What does Δx represent in the context of the difference quotient?
What does Δx represent in the context of the difference quotient?
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Under which condition is a function not differentiable?
Under which condition is a function not differentiable?
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How can the instantaneous rate of change at a specific x value be found?
How can the instantaneous rate of change at a specific x value be found?
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What is true about derivatives?
What is true about derivatives?
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What does a tangent line do to a curve at a given point?
What does a tangent line do to a curve at a given point?
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What condition must be met for a function to be differentiable?
What condition must be met for a function to be differentiable?
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What type of point causes a function to be non-differentiable?
What type of point causes a function to be non-differentiable?
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Using the Power Rule, what is the derivative of the function $f(x) = 5x^3$?
Using the Power Rule, what is the derivative of the function $f(x) = 5x^3$?
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What is the derivative of the function $f(x) = 4x^2$ using the Constant Multiple Rule?
What is the derivative of the function $f(x) = 4x^2$ using the Constant Multiple Rule?
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What does the Quotient Rule state for finding the derivative of $h(x) = \frac{f(x)}{g(x)}$?
What does the Quotient Rule state for finding the derivative of $h(x) = \frac{f(x)}{g(x)}$?
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Which of the following is the correct derivative of the function $f(x) = \sin(x)$?
Which of the following is the correct derivative of the function $f(x) = \sin(x)$?
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What is the derivative of $h(x) = \tan(x)$?
What is the derivative of $h(x) = \tan(x)$?
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What is the derivative of the function $f(x) = 2^x$?
What is the derivative of the function $f(x) = 2^x$?
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What is the derivative of the inverse sine function $y = \sin^{-1}(x)$?
What is the derivative of the inverse sine function $y = \sin^{-1}(x)$?
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Study Notes
Difference Quotient
- The difference quotient is used to calculate the slope of a tangent line to a curve at a point.
- There are two forms:
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f(x + Δx) - f(x) / Δx -
f(x) - f(a) / x - a
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Finding the Slope of a Tangent Line
- The slope of a line tangent to a curve at a point represents the instantaneous rate of change of the curve at that point.
- The tangent line only touches the curve at the given point, unlike a secant line which intersects the curve at two points.
- As the distance between two points on the curve approaches zero, the slope of the secant line approaches the slope of the tangent line.
- The limit as Δx approaches zero of the difference quotient
f(x + Δx) - f(x) / Δxrepresents the slope of the tangent line at a given point.
Δx and the Difference Quotient
- Δx represents the change in the x values or the distance between two x values.
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f(x + Δx)represents the function evaluated atx + Δx, which is the x value plus the distance between the x values.
Derivative
- The derivative is a function that represents the slope of its parent function at a particular point.
- It can be found by evaluating the difference quotient at a specific value of x or by finding the limit of the difference quotient as Δx approaches zero.
Instantaneous Rate of Change
- The instantaneous rate of change is the rate of change of a function at a specific point in time.
- The tangent line represents the instantaneous rate of change at a point.
Evaluating the Difference Quotient
- To evaluate the difference quotient at a specific x value, substitute that x value into the function.
- To find the instantaneous rate of change at a specific x value, evaluate the difference quotient at that x value or find the limit of the difference quotient as Δx approaches zero.
The Derivative
- The derivative of a function represents the slope of the original graph at any given point.
- The derivative can be found using the difference quotient:
- limit as Delta X approaches zero of (f(x + Delta X) - f(x)) / Delta X
- This equation gives you the slope of the tangent line at a given point.
- To find the slope at x = 2, for example, you would plug in 2 for x in the derivative function.
Differentiability
- A function is differentiable if and only if it has a tangent line at every point.
- A function is not differentiable if it is discontinuous.
- A function is not differentiable if it has a corner.
- A corner represents a point on the graph where the slope approaches two different values from the left and from the right.
- A corner does not have to be a 90-degree angle.
- A function is not differentiable if it has a cusp - a special kind of corner.
- A cusp involves a graph where the one-sided limits of the difference quotient approach negative infinity and positive infinity, respectively.
- Since the one-sided limits do not exist, the limit of the difference quotient does not exist, and therefore the function is not differentiable.
- A function is not differentiable if it has a vertical tangent line.
- While the one-sided limits of the difference quotient may be equal to each other, they still equal positive infinity.
- Since the limit does not exist, the function is not differentiable.
- A function is locally linear, which means that if you zoom in far enough on the graph, it will appear linear.
Derivative Rules
- The derivative of a constant, such as y = 4, is always zero because the slope of a horizontal line is always zero. This is true for any constant value.
- The derivative of a first degree function, such as y = ax (where a is a constant), is always equal to a.
- The derivative of a function in the form of ax^n (where a and n are constants) is equal to a * n * x^(n-1).
- The derivative of a constant multiple of a function, such as f(x) = 4x², is equal to the constant multiple outside of the derivative operator.
Summary of Differentiability
- A function is differentiable if it is continuous and has no corners or cusps.
- A function is continuous if it is differentiable.
- The intermediate value theorem applies to derivatives: If there are two points on the derivative function, then the derivative function must achieve every point between those two points.
- The intermediate value theorem is based on the fact that the derivative function is continuous because the original function was differentiable.
- When we solve for the derivative of a function, we find a pattern.
- These patterns form the basis for the derivative rules, which encompass a universal set of rules for taking derivatives.
Derivative Rules
- Power Rule When two like bases are multiplied, add their exponents. This applies to functions with invisible exponents of one, resulting in an *x* raised to the power of (*n* + 1), where *n* is the original exponent.
- Constant Multiple Rule The constant multiple of a function can be factored out of the derivative operation.
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Product Rule Used for finding the derivative of a function *h(x)* that can be represented as the product of two functions, *f(x)* and *g(x)*:
- Formula: *h'(x) = f'(x) * g(x) + g'(x) * f(x)*
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Quotient Rule: Used for finding the derivative of a function *h(x)* that can be represented as the quotient of two functions, *f(x)* and *g(x)*:
- Formula: *h'(x) = (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2*
- Mnemonic for Quotient Rule: "Low D High - High D Low, over Low squared."
Trigonometric Derivatives
- Sine: The derivative of *sin(x)* is *cos(x)*
- Cosine: The derivative of *cos(x)* is *-sin(x)*
- Tangent: The derivative of *tan(x)* is *(sec(x))^2*
- Cosecant: The derivative of *csc(x)* is *-csc(x) * cot(x)*
- Secant: The derivative of *sec(x)* is *sec(x) * tan(x)*
- Cotangent: The derivative of *cot(x)* is *-csc(x)^2*
- Memorization Tip: Derivatives of trig functions starting with 'c' (cosine, cosecant, cotangent) have negative signs.
Inverse Trigonometric Derivatives
- Inverse Sine: The derivative of *sin^-1(x)* is *1 / (√(1 - x^2)) *
- Inverse Cosine: The derivative of *cos^-1(x)* is *-1 / (√(1 - x^2)) *
- Inverse Tangent: The derivative of *tan^-1(x)* is *1 / (1 + x^2) *
- Inverse Cosecant: The derivative of *csc^-1(x)* is *-1 / (|x| * √(x^2 - 1)) *
- Inverse Secant: The derivative of *sec^-1(x)* is *1 / (|x| * √(x^2 - 1)) *
- Inverse Cotangent: The derivative of *cot^-1(x)* is *-1 / (x^2 + 1). *
Exponential and Logarithmic Derivatives
- Logarithmic Function: The derivative of *log_a(x)* is *1 / (x * ln(a)) *
- Natural Logarithm: The derivative of *ln(x)* is *1 / x*
- Exponential Function: The derivative of *a^x* is *a^x * ln(a)*
- Natural Exponential Function: The derivative of *e^x* is *e^x*
- Note: Common mistake: The derivative of a function like *2^x* is *2^x * ln(2)*, not a product rule problem because *ln(2)* is a constant.
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Description
This quiz covers the concept of the difference quotient and its application in finding the slope of a tangent line to a curve. Understand how to use the difference quotient formulas and the significance of Δx in calculus. Test your knowledge on calculating instantaneous rates of change using these principles.
