Calculus Practice Test 5

Show all work and justify each step.

(1) A particle moves along the circle in such a way that, when it is at the point its -coordinate is increasing at the rate of 5 units per second. How fast is the -coordinate changing at that instant?

    (a) 5

    (b)

    (c)

    (d)
    
    (e)


(2) In a healthy person of height inches, the average pulse rate (in beats per minute) is modeled by the formula Estimate the change in pulse rate that corresponds to a height change from in to in.

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    

(3) If when find

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)


(4) Find an equation of the curve that satisfies the following conditions: at each point on the curve, the slope of the tangent line is the curve passes through the point

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)


(5) Find an equation for the perpendicular to at the point

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(6) An object is traveling along a straight line with position at time given by Find the distance travelled over the time interval

    (a) 107
    
    (b) 77
    
    (c) 97
    
    (d) 131
    
    (e) 119


(7) If is differentiable for all values of when and find at

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)


(8) Find the slope of the tangent to the graph of at the point

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(9) If , then is

    (a) 0
    
    (b)
    
    (c)
    
    (d) does not exist
    
    (e) not enough information given
    
    
(10) If , then is

    (a) 0
    
    (b)
    
    (c)
    
    (d) does not exist
    
    (e) not enough information given
    
    
(11) The function is discontinuous

    (a) nowhere
    
    (b) at only
    
    (c) at only
    
    (d) at 2 only
    
    (e) at and only


(12) If , the is discontinuous at

    (a) 0 only

    (b) 1 only
    
    (c) and 1 only
    
    (d) no point of the domain
    
    (e) every point of the domain
    
    
(13) Constants and are chosen such that the function is continuous for all The product is

    (a) 3
    
    (b) 2
    
    (c) 6
    
    (d) 9
    
    (e) undefined


(14) If , then is

    (a) 0
    
    (b)
    
    (c)
    
    (d) 2
    
    (e) does not exist
    

(15) Interpret as a derivative.

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)


(16) Suppose is a function with the property that Find where

    (a)
    
    (b)
    
    (c)
    
    (d) undefined
    
    (e) none of these


(17) Suppose we know that a function has derivative for all and Use a tangent-line approximation to estimate the value of

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) none of these


(18) If and what is

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) none of these


(19) The value of at is

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) does not exist


(20) Give an expression in terms of a limit that gives the area between the graph of the -axis, and the lines and

    (a)
    
    (b)

    (c)
    
    (d)

    (e)
    
    
(21) Two cars start at the same point at noon. One moves east at the rate of 65 mi/hr. The other moves north at the rate of 75 mi/hr. At what rate is the distance between them changing after 2 hours?


(22) Given and show that


(23) Given that compute at the point


(24) Sketch the graph of one function with all the following properties:

    (i) for and
    
    (ii) for and
    
    (iii) the graph has only one critical point
    
    (iv) the graph has no inflection points
    
    (v)
    
    (vi)
    
    (vii)
    
    (viii)    


(25) A Norman window has the shape of a rectangle with a semicircle on top (i.e., the diameter of the semicircle is equal tot he width of the rectangle). If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.

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