Rotated Ellipse Problem: Figure 4-10g shows the graph of 9x2 20xy + 25y2 16x + 10y 50 =

PART 1: No calculators allowed (T1T5) Write the power rule for derivatives. That is, write if y = uv, where u and v are differentiable functions of x.

PART 1: No calculators allowed (T1T5)In deriving the quotient rule for derivatives, where y = u/v, you encountered Show how the quotient rule follows from this complex fraction by multiplying, then taking the limit.

PART 1: No calculators allowed (T1T5)Use the quotient rule and the fact that cot x = (cos x)/(sin x) to prove that x = csc2 x.

PART 1: No calculators allowed (T1T5) Use the definition of the inverse trigonometric functions and appropriate trigonometry and calculus to show that y = sin1 x =

PART 1: No calculators allowed (T1T5)For this function, find and . x = t 2 y = t 4

PART 2: Graphing calculators allowed (T6T24) Is the function c(x) = cot 3x increasing or decreasing when x = 5? At what rate?

PART 2: Graphing calculators allowed (T6T24)If f(x) = sec x, find (2). Show that the difference quotient approaches (2) by making a table of values for each 0.001 unit of x from x = 1.997 through x = 2.003.

PART 2: Graphing calculators allowed (T6T24)Sketch the graph of a function that has all of these features. The value f(2) = 7. The function f(x) increases slowly at x = 2. The value f(1) is positive, but f(x) decreases rapidly at x = 1. The function f(x) is continuous at x = 2 but is not differentiable at that point.

PART 2: Graphing calculators allowed (T6T24) Is the function c(x) = cot 3x increasing or decreasing when x = 5? At what rate Use the most time-efficient method to prove that the general linear function, f(x) = mx + b, is continuous for every value of x = c.

For Problems T10T15, find the derivative.f(x) = sec 5x

For Problems T10T15, find the derivative.y = tan7/3 x

For Problems T10T15, find the derivative. f(x) = (2x 5)6(5x 1)2

For Problems T10T15, find the derivative.f(x) =

For Problems T10T15, find the derivative.x = sec 2t y = tan 2t3

For Problems T10T15, find the derivative. y = 4 sin1 (5x3)

Rotated Ellipse Problem: Figure 4-10g shows the graph of 9x2 20xy + 25y2 16x + 10y 50 = 0 Figure 4-10g The figure is an ellipse that is rotated with respect to the x- and y-axes. Evaluate dy/dx at both points where x = 2. Show that each answer is reasonable

Let f(x) = x3 + 1, if x 1 a(x 2)2 + b, if x > 1 Use one-sided limits to find the values of the constants a and b that make function f differentiable at x = 1. Show that the limits of f(x) as x approaches 1 from the right and from not differentiable at x = 1.

Let y = x7/3. Raise both sides to the third power, getting y3 = x7, and differentiate both sides implicitly with respect to x using the power rule with positive integer exponents. Show that the answer you get for this way is equivalent to the one you get using the power rule directly on y = x7/3. Airplane Problem: For Problems T19T23, an airplane is flying level at 420 mi/h, 5 miles above the ground (Figure 4-10h). Its path will take it directly over an observation station on the ground (see figure). Let x be the horizontal distance, in miles, between the plane and the station, and let be the angle, in radians, to the line of sight.

Show that = cot 1

Find:

If t is time in hours, what does dx/dt equal?

Find d /dt as a function of x. Plot the graph on your grapher and sketch the result.

Where is the plane when is changing fastest?

What did you learn as a result of taking this test that you did not know before?