Calculus I Practice

2311 (Calculus 1) Last Initial:
Calculus I Practice Test 2
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This test has 19 questions. The total number of points is 0.
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1 6 11
2 7 12
3 8 13
4 9 14
5 10 15
Practice Test 2 Calc I – Spring 2019

1. State and prove the Product Rule and the Quotient Rule.
2. Show that d
   dx [tan (x)] = sec2
   (x).
   2
3. Compute d
   dx
   8x
   7
   7
   +
   7x
   6
   6

• 3x
  10 − π
  3
  
  .

1. Let s(u) = 5e
   2u
   3+2u +
   7
   √
   u
   3
   . Compute s
   0
   (u).
2. Let g(θ) = p
   ln(2θ). Compute g
   0
   (θ).
3. Let f(x) = (2x + 5)20(x
   2 + 5x + 3)15. Compute f
   0
   (x).
4. Let h(p) =
   tan (5p +
   10
   p
   )
   20
   . Compute h
   0
   (p).
   3
5. Let s(t) = t
   2 − 5t + 3
   cos (t
   2 + 1). Compute s
   0
   (t).
6. Let f(x) = (e
   x
   2

• 7x + 1)10. Compute f
  0
  (x).

1. Let g(x) = x · ln (x
   2 + 5x + 3). Compute g
   0
   (x).
2. Let f(x) = [tan (x)]x
   . Find f
   0
   (x).
3. Use logarithmic differentiation to show that d
   dx[x
   n
   ] = nxn−1
   .
   4
4. To Infinity and Beyond!
   -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4
   -2
   -1.5
   -1
   -0.5
   0.5
   1
   1.5
   2
   0
   Consider the curve above, described by the equation 8y
   2 + 2x
   2y
   2 + y
   4 = 14x
   2 − 3x
   4
   (a) On the graph, sketch (approximately) the line tangent to the curve at (1, 1).
   (b) Use implicit differentiation to get an expression containing dy
   dx.
   (c) Use your work in part (b) to find the slope of the tangent line at (1, 1).
   (d) Use your work in part (c) to find an equation of the tangent line at (1, 1).
   (e) Is your answer for part (d) consistent with the line you drew for part (a)? Explain why or why not.
   5
5. The position of a particle in time is given by:
   p(t) = t · e
   −t
   ,
   where t is in hours and p(t) is in meters.
   1 2 3 4 5 6
   0.2
   0.4
   t (hours)
   p(t) (meters)
   (a) Find velocity as a function of time.
   (b) Use part (a) to compute the instantaneous velocity when t = 3 hours, and include units.
   (c) Use part (a) to find the time(s) when the velocity is 0.
   (d) Find acceleration as a function of time.
   6
6. Consider the following graph of functions f and g.
   1
   1
   f
   g
   Let F(x) = f(x) · g(x), G(x) = f(x)
   g(x)
   , and H(x) = f(g(x)). Compute F
   0
   (0), G0
   (0), and H0
   (0).
7. Sketch the graph of a function f that satisfies the following conditions:
   • lim x→−∞
   f(x) = 1,
   • f
   0
   (−1) = 2,
   • f
   0
   (1) = 0,
   • f
   0
   (x) > 0 for x > 1, and
   • f(3) = 2.
   7
8. (a) Define linearization at x = a.
   (b) Use linearization to approximate ln (1.01).
9. A spherical snowball is melting under the sun at a constant rate of 4πcm3/s. At what rate is the
   radius decreasing when the radius is 1cm? Show all work and include units in your answer.
10. A rocket is launched and travels straight up at a constant velocity of 750 mph. An observer is 1.5
    miles away watching the launch. How fast is the distance from the observer to the rocket increasing when
    the rocket is 2 miles high? Show all work and include units in your answer.
    8

Sample Solution