1.   Evaluate the integral shown below. (Hint: Try the substitution u = (7x2 + 3). )

2.   Evaluate the integral shown below. (Hint: Apply a property of logarithms first.)

3.   Use the Fundamental Theorem of Calculus to find the derivative shown below.

4.   For the function shown below, sketch a graph of the function, and then find the SMALLEST possible value and the LARGEST possible value for a Riemann sum of the function on the given interval as instructed.

5.   Use L’Hôpital’s Rule to find the limit below.

 

       

6.   Use L’Hôpital’s Rule to find the limit below. (Hint: The indeterminate form is f(x)g(x))

7.   Solve the following problem.

 

      The 9 ft wall shown here stands 30 feet from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

8.   For the function shown below, identify its local and absolute extreme values (if any), saying where they occur.

 

       

 

            The function f(x) is a polynomial and is defined for all real values of x. As a       result, the absolute extreme values will be determined by the end behavior of the

            function. As x -∞, the function value tends toward +∞. Similarly, as x +∞,

            the function value tends toward -∞. As a result, this function has no absolute       extrema.

9.   Find a value for “c” that satisfies the equation  in the conclusion of the Mean Value Theorem for the function and interval shown below.

10. Find the equation of the tangent line to the curve whose function is shown below at the given point.

 

      , tangent at (2, 1)

11. Use implicit differentiation to find .

12. Given y = f(u) and u = g(x), find

 

13. Find y’.

14. Find the derivative of the function “y” shown below.

15. Solve the problem below.

 

      One airplane is approach an airport form the north at 163 km/hr. A second airplane approaches from the east at 261 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 31 km away from the airport and the westbound plane is 18 km from the airport.

 

            Taking north as the positive y direction, and east as the positive x direction, the       velocity of the southbound plane is dy/dt = -163 km/hr, and the velocity of the       westbound plain is dx/dt = -261 km/hr.

 

            With the airport at the origin of the coordinate system, where x is the distance       from the airport to the westbound plane, and y is the distance between the airport

            and the southbound plane, the distance between the two planes is:

 

                         

 

            Differentiating d with respect to t, and recognizing that x  and y are also functions

 

16. Find the intervals on which the function shown below is continuous.

 

       

17. A function f(x), a point c, the limit of f(x) as x approaches c, and a positive number   is given. Find a number  such that for all x,  

19. Solve the “composite function” problem shown below.

      If  and , find . What is ?

20. Find the limit shown below, if it exists

 

       

 

 

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