Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

3. Techniques of Differentiation

Basic Rules of Differentiation

Calculus

3. Techniques of Differentiation

Basic Rules of Differentiation

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2:47 minutes

Problem 27

Textbook Question

Vehicular stopping distance Based on data from the U.S. Bureau of Public Roads, a model for the total stopping distance of a moving car in terms of its speed is s = 1.1v + 0.054v², where s is measured in ft and v in mph. The linear term 1.1v models the distance the car travels during the time the driver perceives a need to stop until the brakes are applied, and the quadratic term 0.054v² models the additional braking distance once they are applied. Find ds/dv at v = 35 and v = 70 mph, and interpret the meaning of the derivative.

Verified step by step guidance

First, identify the function given for the stopping distance: s = 1.1v + 0.054v². This function represents the total stopping distance in terms of the car's speed v.

To find the derivative ds/dv, apply the rules of differentiation to each term in the function. The derivative of a constant times a variable, such as 1.1v, is simply the constant. The derivative of 0.054v² is found using the power rule, which states that d/dx of x^n is n*x^(n-1).

Calculate the derivative of the linear term: d/dv of 1.1v is 1.1.

Calculate the derivative of the quadratic term: d/dv of 0.054v² is 2 * 0.054 * v^(2-1) = 0.108v.

Combine the derivatives to find ds/dv: ds/dv = 1.1 + 0.108v. Evaluate this derivative at v = 35 mph and v = 70 mph to understand how the stopping distance changes with speed. The derivative represents the rate of change of stopping distance with respect to speed, indicating how much the stopping distance increases for each additional mph of speed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function represents the rate at which the function's value changes with respect to changes in its input. In this context, ds/dv is the derivative of the stopping distance s with respect to speed v, indicating how the stopping distance changes as the speed of the car changes. Calculating this derivative helps understand the sensitivity of stopping distance to changes in speed.

Power Rule for Differentiation

The power rule is a basic rule in calculus used to find the derivative of a function of the form f(x) = ax^n. The derivative is given by f'(x) = n*ax^(n-1). For the stopping distance function s = 1.1v + 0.054v², applying the power rule allows us to differentiate each term separately, resulting in ds/dv = 1.1 + 0.108v, which can then be evaluated at specific speeds.

Interpretation of Derivatives in Context

Interpreting the derivative in the context of the problem involves understanding what the calculated rate of change signifies in real-world terms. For the stopping distance model, ds/dv at a specific speed tells us how much the stopping distance is expected to increase for each additional mph of speed. This interpretation is crucial for assessing safety and understanding the impact of speed on stopping distance.

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Related Practice

Textbook Question

Slopes and Tangent Lines

b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope?

Textbook Question

Quadratics having a common tangent line The curves y = x² + ax + b and y = cx − x² have a common tangent line at the point (1,0). Find a, b, and c.

Textbook Question

a. Find an equation for the line that is tangent to the curve y = x³ − 6x² + 5x at the origin.

[Technology Exercise] b. Graph the curve and tangent line together. The tangent line intersects the curve at another point. Use Zoom and Trace to estimate the point’s coordinates.

[Technology Exercise] c. Confirm your estimates of the coordinates of the second intersection point by solving the equations for the curve and tangent line simultaneously.

Textbook Question

Additional Applications

Bacterium population

When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time t (hours) was b = 10⁶ + 10⁴t − 10³t². Find the growth rates at

a. t = 0 hours.

b. t = 5 hours.

c. t = 10 hours.

Textbook Question

Airplane takeoff Suppose that the distance an aircraft travels along a runway before takeoff is given by D = (10/9)t², where D is measured in meters from the starting point and t is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 200 km/h. How long will it take to become airborne, and what distance will it travel in that time?

Multiple Choice

Find the indicated derivative.

$f(x)=3x-4$