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

2. Intro to Derivatives

Tangent Lines and Derivatives

Calculus

2. Intro to Derivatives

Tangent Lines and Derivatives

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2:58 minutes

Problem 3.72

Textbook Question

Are there any points on the curve y = x - 1/(2x) where the slope is 2? If so, find them.

Verified step by step guidance

To find the points on the curve where the slope is 2, we need to first determine the derivative of the function y = x - \(\frac{1}{2x}\). The derivative, y', represents the slope of the tangent line at any point on the curve.

Differentiate the function y = x - \(\frac{1}{2x}\) with respect to x. The derivative of x is 1, and the derivative of \(-\frac{1}{2x}\) can be found using the power rule and chain rule. Rewrite \(-\frac{1}{2x}\) as \(-\frac{1}{2}x^{-1}\) and differentiate.

The derivative of \(-\frac{1}{2}x^{-1}\) is \(\frac{1}{2}x^{-2}\) or \(\frac{1}{2x^2}\). Therefore, the derivative of the function y = x - \(\frac{1}{2x}\) is y' = 1 + \(\frac{1}{2x^2}\).

Set the derivative equal to 2 to find the x-values where the slope of the tangent is 2: 1 + \(\frac{1}{2x^2}\) = 2.

Solve the equation \(\frac{1}{2x^2}\) = 1 for x. This will give you the x-values where the slope is 2. Substitute these x-values back into the original equation y = x - \(\frac{1}{2x}\) to find the corresponding y-values, thus identifying the points on the curve.

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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 measures the rate at which the function's value changes as its input changes. It is often interpreted as the slope of the tangent line to the curve at a given point. For the curve y = x - 1/(2x), finding the derivative will help determine where the slope equals 2.

Finding Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are essential for analyzing the behavior of the function, including identifying local maxima, minima, and points of inflection. In this context, we need to set the derivative equal to 2 to find specific points on the curve.

Solving Equations

Solving equations involves finding the values of variables that satisfy a given mathematical statement. In this case, after determining the derivative and setting it equal to 2, we will solve for x to find the corresponding y-values on the curve. This process is crucial for identifying the points where the slope of the curve is exactly 2.

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Textbook Question

The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>

Explain why the average rate of growth from 1950 to 1960 is a good approximation to the (instantaneous) rate of growth in 1955.

Textbook Question

Estimate the instantaneous rate of growth in 1985.

Textbook Question

72–76. Tangent lines Find an equation of the line tangent to each of the following curves at the given point.

y = 3x³+ sin x; (0, 0)

Textbook Question

79–82. {Use of Tech} Visualizing tangent and normal lines

b. Graph the tangent and normal lines on the given graph.

(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)

Textbook Question

Find the points on the curve y = 2x³ - 3x² - 12x + 20 where the tangent line is

a. perpendicular to the line y = 1 - (x/24).

b. parallel to the line y = √2 - 12x.

Textbook Question

Find the points on the curve y = tan x, -π/2 < x < π/2, where the normal line is parallel to the line y = -x/2. Sketch the curve and normal lines together, labeling each with its equation.

Textbook Question

For what value of c is the curve y = c/ (x + 1) tangent to the line through the points (0, 3) and (5, -2)?

Textbook Question

In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.

f(x) = √(x + 1), (8, 3)

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