Jose San State University Math 30 - Fall 09
Final Exam with Solutions
Show Your Work Simplify Your Answers
Name:
Problem 1: f g h(x). Solution:
If f (x) =
x - 3, g(x) = x2 , and h(x) = x3 + 2, find
x6 + 4x3 + 1
Problem 2: Solution: 3/5
x2 + 5x + 4 Evaluate lim 2 , if it exists. x-4 x + 3x - 4
Problem 3: Solution: 2
Find lim
4u4 + 5 . u (u2 - 2)(2u2 - 1)
Problem 4: Solution:
Find the derivative of g(x) = cosh(ln x).
1 sinh(ln x) x
Problem 5:
Differentiate f (x) =
1 x+ 3 x
2
.
Solution: 2(x1/2 +x-1/3 )(1/2x-1/2 -1/3x-4/3 ) = 1+1/3x-5/6 -2/3x-5/3
Problem 6: Solution:
Find f (x) and f (x), for f (x) = x4 ex .
f (x) = ex (x4 + 4x3 ); f (x) = e x (x4 + 8x3 + 12x2 )
Problem 7: Differentiate f (x) = sin2 (esin x ). Solution: 4 sin x cos x esin
2
2
x
sin(esin x ) cos(esin x )
2
2
Problem 8: Find the derivative of f (t) = tan(et ) + etan t . Solution: et sec2 (et ) + sec2 tetan t
Problem 9: Find dy/dx given that x2 y 2 + x sin y = 4. sin y + 2xy 2 Solution: dy/dx = - 2yx2 + x cos y
Problem 10: Differentiate y = ( x)x . Solution: y = ( x)x (1/2 + ln x)
Problem 11: A ladder 15 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 5 ft/s, how fast is the top of the ladder sliding down the wall the when bottom of the ladder is 10 ft from the wall? Solution: -10/ 5 ft/sec
Problem 12: Find the linearization L(x) of the function at a = 1. Use L(x) to approximate the number 3.98. Solution: 1 7 x + ; 1.995 4 4
f (x) =
x+3
Problem 13: Let f (x) = 2 + 3x - x3 . (a) Find the local maximum and minimum values of f . (b) Find the coordinates (x, y) of the inflection point(s) of f . Solution: (a) 4, 0; (b) (0, 2)
Problem 14: Evaluate lim Solution: 1
ln x . x1 x - 1
Problem 15: Evaluate lim Solution: 0
1 - cos x . x0 x2 + x
Problem 16: Find the coordinates (x, y) of the point on the line y = 10x+2 that is closest to the point (1, 5). Solution: (31/101, 512/101)
Problem 17: Find two positive numbers whose product is 100 and whose sum is a minimum. Solution: x = 10, y = 10
Problem 18: Newton's method to find the second approximation (x2 ) Use 3 to the number 3. (Hint. Use 3x1 = 1 as the first approximation. Notice that 3 3 is a root of the equation x - 3 = 0.) Solution: x2 = 5/3
Problem 19: Find the most general antiderivative of the function f (x) = x(2 - x)2 . Solution: x4 4x3 - + 2x2 + C 4 3
Problem 20: Find the absolute maximum and absolute minimum values of f (x) = x3 - 3x + 1 on the closed interval [0, 3]. Solution: Absolute max = 19, absolute min = -1