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Math 125 Exam Solutions - Prof. Brick, Exams of Calculus

Solutions to exam 1, exam 2, exam 3, and the final exam for math 125 with prof. Brick. The problems involve calculus concepts such as limits, derivatives, integrals, and optimization.

Typology: Exams

2012/2013

Uploaded on 03/31/2013

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Fall 98 Math 125 Exam 1 Prof. Brick
Do the problems in order in your bluebook. Show your work.
1. For what value of kis the following function continuous at x=1?
f(x)=1
2x2+k, if x≤1;
x, if x>1.
2. Use a linear approximation for √xnear x= 16 to approximate √16.3
3. Find all xwhere the function y=2x3−3x2is increasing at a faster and faster rate.
4. Find the equation of the line tangent to f(x)=√x−64
x2at x=4
5. A ball is thrown up from a window. Its height after tseconds is s(t)=96+16t−16t2.
At what velocity does it hit the ground ?
6. Find the derivative of h(t)= 3
√t2+e2−et+3
t4+π3(you may use the shortcuts).
7. Find lim
x→+∞
5x−9x6
2x6−17 .
8. Use the definition (with limits) to find the derivative of f(x)= 1
7x+1
pf3
pf4

Partial preview of the text

Download Math 125 Exam Solutions - Prof. Brick and more Exams Calculus in PDF only on Docsity!

Do the problems in order in your bluebook. Show your work.

  1. For what value of k is the following function continuous at x = 1?

f (x) =

2 x

(^2) + k, if x ≤ 1; x, if x > 1.

  1. Use a linear approximation for

x near x = 16 to approximate

  1. Find all x where the function y = 2x^3 − 3 x^2 is increasing at a faster and faster rate.
  2. Find the equation of the line tangent to f (x) =

x −

x^2

at x = 4

  1. A ball is thrown up from a window. Its height after t seconds is s(t) = 96 + 16t − 16 t^2. At what velocity does it hit the ground?
  2. Find the derivative of h(t) = 3

t^2 + e^2 − et^ +

t^4

  • π^3 (you may use the shortcuts).
  1. Find lim x→+∞

5 x − 9 x^6 2 x^6 − 17

  1. Use the definition (with limits) to find the derivative of f (x) =

7 x + 1

Do the problems in order in your bluebook. Use algebraic techniques.

  1. Find the line tangent to x^3 y^2 − 9 x^4 = tan(3x^2 − y) at (1, 3)
  2. Use the first derivative test to find the local min/max’s of y = x · (x^2 − 1) (^23)
  3. Find the inflection points of y = ln(x^2 + 1). Also find where y is concave up or down.
  4. Use the second derivative test to classify the the local min/max’s of y = ex

(^3) −x .

  1. A kite is flown at a constant height of 210 feet. Its speed is 7π ft/sec. At what rate in degrees/sec is the angle made by the string (with the horizontal) changing when 350 ft of string has been let out?
  2. A man 6 feet tall is walking away at the constant speed of 12 feet/sec from a streetlight that is 18 feet high. At what rate is the length of his shadow changing?
  3. Find the absolute min/max’s of y = sin(x) + cos(x) over [3, 5]
  4. Find the limit lim x→ 0

ex^ − x − 1 1 + x^2 − cos(x)

Fall 98 Math 125 Final Exam Prof. Brick

Do the problems in order in your bluebook. Use algebraic methods

  1. Let cost be given by C(x) = 1000 + 40x and the demand curve be given by the formula p(x) = 80 − x (where p is the price and x is the number sold). Maximize the profit. [Note: there is a typo, cost should be 100 + 40x]
  2. Find the equation of the line tangent to f (x) = xx^ at x = 2
  3. Approximate

8 .9 using a linear approximation of

x.

  1. Use the definition of the derivative to find y′^ where y =

x^2 + 1.

  1. Use the first derivative test to find the local min/max’s of y = (x^2 − 9 x)

(^23) .

  1. Use the 2nd derivative test to classify the the local min/max’s of g(x) = arctan(x^2 + 1).
  2. A thirteen foot ladder is leaning up against a wall. The end of the wall slips and slides down at a rate of 2ft/sec. How fast is the other end moving away from the wall, when the top of the ladder is 5 feet off the ground?
  3. Find the absolute min/max’s of y = 2x^3 + 3x^2 − 12 x + 10 over [0, 2].
  4. Find lim x→ 0

ex^ − x − 1 x^2

  1. A box is to be constructed by cutting out square corners from a piece of cardboard measuring 8 12 inches by 11 inches. What size corner yields maximum volume for the box?
  2. Find

sin^4 (x) cos^3 (x) dx

  1. Find

∫ (^) e 3

e

e x ln(x)

dx