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Exam 2: The Derivative

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Question 21

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Find f '(x) if f(x) = x⁴ cos x.

A) 4x³ cos x - x⁴ sin x
B) 4x³ cos x
C) 4x³ sin x
D) -4x³ cos x
E) -4x³ sin x

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Question 22

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Find f '( $\theta$ ) if .

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Question 23

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Find if .

Find f '(x) if .

Let f(x) = ax³ + b, where a and b are constants. Use the method of Section 3.1 to show that the slope of the tangent to the graph of f at x = x₀ is .

Answer true or false. is differentiable everywhere.

Given that f(1) = 5 and f '(1) = 6, find an equation for the tangent line to the graph of y = f(x) at the point where x = 1.

Find y''(x) if .

Find equations for the tangents and normals to the graph of y = 30 - x - x² at the points where the curve intersects the x-axis.

Find f '(x) if .

Find f '(x) if f(x) = 5x⁴.

If y = x⁷ cos x, find dy/dx.

Find dy/dx if .

Find the value of the constant A in y = A sin 3t so that d²y/dt² + 7y = sin 3t.

Show that is continuous but not differentiable at x = 1.

Let . Find the average rate of change of y with respect to x over the interval [4,7].

Answer true or false. If y = sin x, d²y/dx² = sin x.

Given that f(1) = 5, f '(1) = 4 and g(x) = (f(x))^-4, find g '(1).

Answer true or false. y = y''' + 5y' - 2 is satisfied by y = x.

Find f '(x) where f(x) = cos(tan 4x).