2. Intro to Derivatives

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).

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b. parallel to the line y = √2 - 12x.

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.

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

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)

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.

h(t) = t³ + 3t, (1, 4)

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.

g(x) = 8 / x², (2, 2)

In Exercises 19–22, find the slope of the curve at the point indicated.

y = x³ − 2x + 7, x = −2

Interpreting Derivative Values

Growth of yeast cells In a controlled laboratory experiment, yeast cells are grown in an automated cell culture system that counts the number P of cells present at hourly intervals. The number after t hours is shown in the accompanying figure.

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a. Explain what is meant by the derivative P'(5). What are its units?

Interpreting Derivative Values

Effectiveness of a drug On a scale from 0 to 1, the effectiveness E of a pain-killing drug t hours after entering the bloodstream is displayed in the accompanying figure.

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a. At what times does the effectiveness appear to be increasing? What is true about the derivative at those times?

Find an equation of the straight line having slope 1/4 that is tangent to the curve y = √x.

Rates of Change

Speed of a rocket At t sec after liftoff, the height of a rocket is 3t² ft. How fast is the rocket climbing 10 sec after liftoff?

Show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b).

[Technology Exercise]

Graph the curves in Exercises 39–48.

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38.

y = 4x²/⁵ − 2x

Tangent line to y = √x Does any tangent line to the curve y = √x cross the x-axis at x = −1? If so, find an equation for the line and the point of tangency. If not, why not?

In Exercises 19–22, find the slope of the curve at the point indicated.

y = (x − 1) / (x + 1), x = 0