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Math 105 B (11 am), C (noon) Fall 2004 FINAL EXAM PUT YOUR NAME ON THE TOP OF EACH PAGE and CIRCLE WHICH SECTION YOU ARE IN. Please write all answers in the space provided on the exam. If you need more space, use the back, but INDICATE CLEARLY to the grader that there is work on the back, and label clearly. You may use scrap paper, but copy all relevant work out neatly on the exam. No partial credit if you don’t show your work.
- 50 points All parts below refer to the function f (x) = sinx^ x.
a. What is the average rate of change of f on [2, 4]?
b. What is the average value of f on [2, 4]?
c. What is the derivative of f at x = 2?
d. Find the equation of the tangent line to f at x = 2.
e. Use local linearization to estimate sin 2. 1 / 2 .1.
f. Do you expect your estimate to be above or below the actual value of sin 2. 1 / 2 .1? Explain.
g. What is limx→ 0 f (x)?
h. Define a function g(x) as
g(x) =
{ f (x) if x 6 = 0 1 otherwise
Is g(x) continuous at x = 0? Explain.
i. Define a function G(x) as
G(x) =
∫ (^) x
0
f (t)dt
Find G(1).
j. Using the same function G(x) defined above, what is G′(x)?
- 10 points Find the solution of the following initial value problem:
dy dx
= 6x^2 + 4x, y(1) = 10
- 10 points Find (^) dxdy for each of the following:
a. y = 5x^2 + πx^ + e−x
b. y =
e^2 − x^2
c. y^3 + yx^2 + x^2 = 3y^2
d. y = tan(arctan(kx))
e. y = ln(
x) + arctan(x^2 )
- 10 points Evaluate the following. (Do all of these by hand, and show work. Do NOT use calculators except possibly to check your answers.)
a.
∫ (^3) x dx
b.
∫ (^7) − 3 x 2 5 x^2 dx
c.
∫ (
x + (^) x^15 + 6x) dx
d.
∫ (^) π 0 (sin^ t^ + cos^ t)^ dt
e. (^) dxd
∫ (^) x 1 cos^
t dt
