Roots (Zeros)


a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.


iv. y = x³ − 33x² + 216x = x(x - 9)(x − 24)

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.

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Find the derivatives with respect to x of the following combinations at the given value of x.

f. (x¹¹ + f(x))⁻², x = 1

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.

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Find the derivatives with respect to x of the following combinations at the given value of x.

g. f(x + g(x)), x = 0

Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.

Find dy/dt when x = 1 if y = x² + 7x − 5 and dx/dt = ¹/₃.

b. Slopes on a tangent curve What is the smallest value the slope of the curve can ever have on the interval −2 < x < 2? Give reasons for your answer.

Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x is

y = 37 sin[(2π/365)(x − 101)] + 25

and is graphed in the accompanying figure.

a. On what day is the temperature increasing the fastest?

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Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x is

y = 37 sin[(2π/365)(x − 101)] + 25

and is graphed in the accompanying figure.

b. About how many degrees per day is the temperature increasing when it is increasing at its fastest?

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Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².