Math 160 Exam III Practice

1) Find an equation of a hyperbola with a vertex (2,2), a focus (3,2) center (1,2). 2) ( 5 points) A searchlight is shaped like a paraboloid. If the light source is located 3 feet from the base along the axis of symmetry and the opening is 6 feet across, how deep should the searchlight be? 3) Graph 4) a) Find an equation for the ellipse that satisfies the following: its center is at (2,2), a focus is located at (2,0) a vertex at (2,5).

5) Find an equation of a parabola with vertex (-2,3) and focus (-3,3). 6) Find an equation of a hyperbola with vertices (4,1), (8,1) and a focus (3,1) 7) Find a partial fraction decomposition of 8) (5 points each) a) b) c) Use notation to express using a starting summation index of 1. D) Compute E) Compute 9) Use induction to show that is divisible by 3. 10)( 5 points each) a) Find the angle between and b) Decompose into two components: the component parallel to and perpendicular to . c) Find the unit vector in the direction of d) Find the vector that has initial point P(1.3) and terminal point Q(-2,1).Math 16

11) Solve 12)Find the partial fraction decomposition of 13)Graph . You must find the vertices and foci. 14) a) Prove that the basic form of a parabola opens to the right can be given as . b) Find an equation of the parabola with directrix and focus 15)A) Decompose into two parts: the projection of u onto v and a vector orthogonal to v, where u=<4,1> and v =<1,5> c) Find the unit vector in the direction of 16) a) Compute b) Compute c) Write the following using the summation symbol

d) Find the 100th term of the sequence e) Find an expression involving an exponent for the 16th term of the sequence f) Compute the term in the expansion of 17) Solve 18)Graph . You must first find foci, center, vertices, and the oblique 19)Sketch the graph . You must first find the directrix and focus. 20)Find a partial fraction decomposition of 21)Find the a) component of parallel to (projection) and b) orthogonal to, where and .

22) a. ( 5 points) Find the angle between and . Round your answer to 1 decimal place

b. Find the unit vector in the direction of c. (5 points) Find the sum of d. ( 5 points) Find the vector , where = (1,4) and =(5,9) e. Find an equation of the ellipse with vertices at (3,5) and (3,11) and a focus at (3,10) 23)A small plane is heading from Southwest to Northeast (that is,)at a speed of 280 miles per hour. The wind is blowing from South to North (what is ) at 40 miles per hour. Find the actual speed and the direction of the plane. 24) Find the term containing in the simplified expansion for 25) a) Compute b) Use sigma notation to write c) Find the sum of the series d) Find the sum of the first 100 integers. e) Find and of an arithmetic series if f) Find and of a geometric sequence if g) Find 26) Use induction to prove 27) Use induction to show 28) Use induction to prove 29) Multiply 30) a) Use sigma notation to write b) Find the sum of the series c) Find the sum of the series d) Find the sum of the first 100 terms of the arithmetic series if e) Use arithmetic formulas to compute f) Find when g) Find and of a geometric sequence if . You may assume that the common ratio is

positive.

31) Sketch the graph of f and determine whether is continuous at x=1 using the definition of continuity. If it is not continuous at x=1, identify the type of discontinuity. 32) Use the definition to compute the derivative of 33) Use definition to compute the derivative for each of the following: a. b)

34) Compute each the following : a) B) C) D) E) F)

35) Let Determine whether f is continuous at 1. If not continuous, identify the type of discontinuity. 36) A given object of mass 5kg is suspended by two cables shown below. Find the magnitude of the tension in each cable.