Book cover for Calculus

Calculus

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

ISBN #9781119379331

7th Edition

5,101 Questions

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33,688 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the power and polynomial differentiation techniques, including the sum/difference rules, the power rule, and the use of the definition of the derivative. It shows how to differentiate functions with positive, negative, and fractional powers and introduces the tangent line approximation for estimating function values. The concepts are grounded in clear examples from physical applications, reinforcing the importance of these differentiation shortcuts in both theoretical and practical contexts.

Learning Objectives

1

Explain and apply the derivative rules for sums, differences, and constant multiples of functions.

2

Understand and utilize the power rule for differentiating polynomials and power functions, including negative and fractional exponents.

3

Demonstrate the use of the definition of the derivative to justify derivative formulas.

4

Apply the tangent line approximation to estimate function values near a given point.

5

Interpret the meaning of the derivative in real-world examples such as wind power generation and motion.

Key Concepts

CONCEPT

DEFINITION

Derivative of Sum and Difference

If f and g are differentiable, then d/dx[f(x) + g(x)] = f'(x) + g'(x) and d/dx[f(x) - g(x)] = f'(x) - g'(x).

Power Rule

For any constant real number n, d/dx(x^n) = n*x^(n-1). This rule applies to positive, negative, and fractional exponents.

Tangent Line Approximation

For values of x near a point a, f(x) is approximately equal to f(a) + f'(a)(x - a). This uses the derivative at a to estimate changes in the function.

Constant Multiple Rule

The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

Definition of the Derivative

f'(x) = lim (h ā†’ 0) [f(x+h) - f(x)] / h, which forms the basis for deriving all differentiation rules.

Example Problems

Example 1

Let $f(x)=7 .$ Using the definition of the derivative, show that $f^{\prime}(x)=0$ for all values of $x$.

Example 2

Let $f(x)=17 x+11 .$ Use the definition of the derivative to calculate $f^{\prime}(x)$.

Example 3

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why. $$y=3^{x}$$

Example 4

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why. $$y=x^{3}$$

Example 5

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why. $$y=x^{\pi}$$
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Step-by-Step Explanations

QUESTION

How do you differentiate f(x) = x^3 using the power rule?

STEP-BY-STEP ANSWER:

Step 1: Identify the power: for f(x) = x^3, n = 3.
Step 2: Apply the power rule: Multiply by the exponent and reduce the exponent by one.
Step 3: Compute the derivative: f'(x) = 3*x^(3-1) = 3x^2.
Final Answer: f'(x) = 3x^2.

Using the Power Rule for f(x) = x^n

QUESTION

Show that the derivative of f(x) = x^(-2) is -2x^(-3) using the definition of the derivative.

STEP-BY-STEP ANSWER:

Step 1: Write f(x+h) = (x+h)^(-2) and f(x) = x^(-2).
Step 2: Form the difference quotient: [(x+h)^(-2) - x^(-2)] / h.
Step 3: Use algebraic manipulation (common denominator, factoring out h) and take the limit as h ā†’ 0.
Step 4: The limit simplifies to -2x^(-3).
Final Answer: f'(x) = -2x^(-3).

Using the Definition of the Derivative for f(x) = x^(-2)

QUESTION

Approximate the power generated by a wind turbine using the tangent line approximation when the wind speed changes slightly.

STEP-BY-STEP ANSWER:

Step 1: Given the power function P = k*v^3 and a known point (vā‚€, Pā‚€), compute the derivative P' = 3k*vā‚€^2.
Step 2: Write the tangent line formula: P ā‰ˆ Pā‚€ + P'(v - vā‚€).
Step 3: Substitute the values at vā‚€ (for example, vā‚€ = 10 m/s, k = 2) to get Pā‚€ = 2000 and P' = 600.
Step 4: Use the formula to estimate P for a nearby value of v (e.g., v = 12 m/s gives P ā‰ˆ 2000 + 600(2) = 3200).
Final Answer: The tangent line approximation yields P ā‰ˆ 3200 kW when v = 12 m/s.

Tangent Line Approximation for a Wind Turbine Power Function

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Common Mistakes

  • Confusing the sum rule by attempting to differentiate term-by-term incorrectly, such as forgetting to differentiate constants (which should yield zero).
  • Misapplying the power rule when dealing with negative or fractional exponents, often leading to sign errors.
  • Ignoring the proper manipulation in the limit process when deriving a derivative from first principles.
  • Using the tangent line approximation outside the range where the linear approximation is valid.
  • Overlooking the fact that a constant derivative (such as for a linear function) implies a constant rate of change.