Exam 2: The Derivative

A particle moves in a straight line from its initial position so that after t seconds, its distance is given by s = t² + t feet from its initial position. Find the average velocity of the particle over the interval [3,5] seconds. Use limits to find the instantaneous velocity of the particle at t = 1 second.

Find f '(t) if f(t) = 4t⁶ - 8.

Use a graphing utility to graph y = t² - 5t + 4 on [0, 10]. If this graph represents a position versus time curve for a particle, the instantaneous velocity of the particle is zero at what time? Assume time is in seconds.

Answer true or false. If , .

Use the definition of the derivative to calculate f '(x) if and find the equations of the tangent line and the normal line to the graph of f at x = 12.

Show by use of a trigonometric identity that .

Find equations for the tangents to the graph of y = 35 - 2x - x² at those points where the curve intersects the x-axis.

If y = x⁴ - 4x + 5, find .

Let f(x) = ax² + b, where a and b are constant. Use the method of Section 3.1 to show that the slope of the tangent to the graph of f at x = x₀ is 2ax₀.

Find d ⁷⁵ of cos x/dx⁷⁵.

Find the instantaneous rate of change of y = 11x⁵ with respect to x at a general point x₀.

Find if y = (x² - 1)(x³ + 3x).

Find dy/dx if y = 3(x⁶ - 6x + 1).

Let . Find the instantaneous rate of change of y with respect to x at the point x = 7.

Let f(x) = cos x. Estimate f '(1) using a graphing utility.

Use the definition of the derivative to calculate f '(x) if f(x) = 2x² - 5x and find the equation of the tangent to the graph of f at x = 2.

Find equations for the tangent and normal lines to the graph of f(x) = sin(16 - x²) at x = 4.

Find f '(x) where .

If , then

If f(4) = 7, f '(4) = 2, g(4) = 2, g '(4) = -2, find F'(4) if F(x) = .