Page 1
Math 151-copyright Joe Kahlig, 09B
1. (18 points) Find
(a) y =
dy
for the following.
dx
√
4
x2 − 7x
(b) y 5 + x2 y 3 = 5x
(c) y = tan(5x2 ) + csc(x)
2. (4 points) Evaluate.
3(2 log3 (4)+3) =
3. (6 points) Find the equation of the tangent line to y = sec(x) − 2 cos(x) at x = π/3
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Page 2
Math 151-copyright Joe Kahlig, 09B
4. (12 points) Compute these limits. Give exact values of these limits.
1
(a) lim− 2 x−3 =
x→3
(b) lim
x→0
10 sin(5x)
=
tan(6x)
3e2x + 4e−2x
=
x→∞ 5e−2x − 7ex
(c) lim
5. (5 points) Find the inverse of the function y = 5 − 2x3
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Page 3
Math 151-copyright Joe Kahlig, 09B
6. (8 points)
(a) Find the linearization, L(x), of the function f (x) =
(b) Use the answer in part (a) to estimate
√
3
x + 60 at a = 4.
√
3
70.
7. (10 points) Solve the following problems for x. Give exact answers when possible.
(a) 7ex−8 = 35
(b) log2 (1 − 2x) + log2 (4x) = 2
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Page 4
Math 151-copyright Joe Kahlig, 09B
8. (6 points) Find f ′′ (x) for f (x) = e3x
2
9. (12 points) The curve is defined by
x = 2t3 − 3t2 − 12t
y = t2 − t + 1
(a) Find all the values of t for which the tangent line is horizontal.
(b) Find all the values of t for which the tangent line is vertical.
(c) Find
dy
evaluated at the point (−13, 1).
dx
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Page 5
Math 151-copyright Joe Kahlig, 09B
10. (5 points) f (x) = 4 + x + ex−2 is a one to one function with inverse g(x). Compute g ′ (7).
11. (5 points) The circumference of a sphere (the length of its ”equatorial circle”) was measured to be 20cm
with a possible error of 1 cm. Estimate a maximum error in the calculated volume of the sphere using
4
differentials. Recall that the volume of a sphere is V = πr3 and the circumference of a circle is C = 2πr.
3
Here r is the radius.
12. (9 points) A trough is 10 feet long and its ends have the shape of isosceles triangles that are 4 feet across
the top and have a height of 1 foot. If the height of the water is decreasing at a rate of 0.15 ft/min when
the water is 6 inches deep, find the rate of change of the volume of the tank at this time.
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