CL1 - Stories: MathExcel: Calculus Among Friends



 
 
   
   
 
   
   
 
 
 
 
 
   
 
   
   


 
 
 
   
 
 
   
 
 
 
 
MathExcel: Calculus Among Friends
  - by Mike Freeman

Sample Worksheet 1

MathExcel, Section 21
Name: _______________________
Worksheet 28
November 9, 1998

Groups should work cooperatively to complete as many of the following as possible. Any questions on problems should be asked by the group, not individuals.

  1. Find all of the horizontal and vertical tangent lines to the following:
    1. x2 + y2 = 2
    2. 3x2 + y2 = 12x

  2. Show that the normal line to the circle x2+y2 = a2 at any point (s,t) passes through the origin. (The normal at any point on the curve is the line perpendicular to the tangent line.)

  3. Show that the curves given by x2 + y2 = 4 and x - 2y = 0 are orthogonal. (Two curves are orthogonal when their tangent lines are perpindicular everywhere the curves interesect.)

  4. Find an equation of the tangent line of the curve y2 = x3+3x2 at the point (1,-2). At what points does this curve have a horizontal tangent line? At what points does this curve have a vertical tangent line? After finding this points, use your calculator to graph the curve and check if your answers make sense. (Hint: write the curve as two functions and graph both functions at the same time.) Sketch the graph on your paper.

  5. Find [(d2x)/( dy2)] if xy = cosy +2x

  6. Find [dy/ dx]:
    1. x = cos(xy)
    2. y = tan23x

  7. Find the slope of the tangent line and the normal line of x2+4xy+y2+3 = 0 at the point (2, -1).

  8. Find the slope of the tangent line to the graph of the equation xy + sin(py) = x3 at the point (0,1).

  9. Use implicit differentiation to determine the following derivatives. Answer in terms of x.

    1. [d/ dx] arcsinx
    2. [d/ dx] arctanx
    3. [d/ dx] arcsec x
    4. [d/ dx] arccsc x

  10. Differentiate the following functions.

    1. f(x) = arctan(2x)
    2. h(t) = Ö{arcsint}
    3. f(y) = earctany
    4. j(x) = x2arccsc(Öx)

  11. The derivative of an inverse of a function can be found by using the formula
    (f-1)¢(x) = [1/( f¢(f-1(x)))]

    Prove this formula in two ways:

    1. By a property of inverses f(f-1)(x) = x. Differentiate both sides of this equation and then solve for (f-1)¢(x).
    2. Let y = f-1(x). Then, by the definition of an inverse function, f(y) = x. Differentiate this function implicitly, and when you have solved for [dy/ dx], you will have proven the formula.

  12. If f(x) = ex, then f-1(x) = ln(x). Use this an the formula above to find the derivative of ln(x).

  13. Verify the formula [d/ dx](ax) = ax ln(a) using that [d/ dx](ex) = ex, chain rule and properties of logs.

  14. Use implicit differentiation, the inverse formula or the log change of base formula to find the derivative of y = logb(x).

  15. Differentiate the following functions:
    1. f(x) = ln(1+x2)
    2. g(t) = log3(x2+1)
    3. h(x) = sin(4x)ln(3x)
    4. g(t) = e3x ln(15)

  16. Differentiate the following. (Hint: use properties of logs to simplify the problem BEFORE taking the derivative.)
    1. g(x) = ln(x4 (x3-4)2 (4x-5))
    2. h(x) = 3ln( [(3-x7)/( (x-2)3 (64+3x)9)]
    3. k(x) = ln(sin(x) cos(x) (3x-4)4)

File translated from TEX by TTH, version 2.25.
On 9 Sep 1999, 15:23.

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