Mathematics 53
Final Exam Exercise Set
I. TRUE or FALSE. 1. The function f (x) = 4x3 − 12x2 has a horizontal tangent line at the point (2, −16). 2. If f 00 (a) = 0, then f has an inflection point at x = a. 3. The graph of the function g(x) = cosh x + sinh x is symmetric with respect to the y-axis. 4. If f is differentiable on [a, b], then f is integrable on [a, b]. 5−x 5. The function f (x) = ln is defined for all x < 5. 5+x 6. If f is continuous at x = a, then f 0 (a) exists. 7. If a function is not differentiable at x0 , then the graph of f has no tangent line at the point (x0 , f (x0 )). 8. If f 0 (a) = 0, then f has a relative extrema at x = a. 9. The exponential function f (x) = ex grows faster than the power function g(x) = xn , for any n. 10. For a positive continuous function f , there is a c ∈ [a, b] such that the rectangle with base [a, b] and height f (c) has the same area as the region under the graph of f from a to b. 11. The hyperbolic cotangent function is an even function. 12. The set of real numbers R is the range of the inverse sine function. 13. If a function f is discontinuous on an interval [a, b], then f is not differentiable on [a, b]. 14. If a function f is integrable on [a, b], then f is continuous on [a, b]. Rx 15. If g(x) is an antiderivative of f (x) on [a, b], then g(x) = a f (t)dt + C, where C is a constant.
II. Discuss continuity of the following functions. cos x, x ln x 1. f (x) = , 1−x −1,
x=0 0<x<1
x ≤ −1 x > −1
x ≤ −4 sin(πx), √ 3. f (x) = x + 5 − 2 , x > −4 x+1
x=1
dy III. Solve for . dx s 4x coth (e−x ) 1. y = 5 sin (x2 ln x) 2 2. ey = cosh (ex + log3 y) − 2x
3. 4tan(xy) = log5 (cosh−1 y) 3 cos 2x
4. y = (1 − cschx )
2 2 − x , 2. f (x) = 1 − √x + 1 , 2x2 + x
5. sech(x2 y) +
6. 3 =
p
5x sin−1
√ = log2 (cosh−1 y) x
x2 − y cos y
2x + 1 7. y = ln 4x x−2
IV. Evaluate the following limits. 1 1 1. lim+ − ln x cosh−1 x x→1
4. lim+ (sinh−1 x)tan x x→0
5. lim+ (cosh x)csc x 2. lim (ex + sinh x)
x→0
cot x
x→0+
6. √
3.
lim
x→+∞
x2 + 2x − 1 3x − 2x2
7.
9x2 + 4 x→−∞ 5 − 6x lim
lim (1 + sin 2x)tan x
+ x→ π 2
V. Evaluate the following integrals. π 2
Z 1. π 6
Z 2. Z 3. Z 4. Z 5. 0
6. 7.
dx √ x + 4 x + 13
8.
π 4
Z
1 + cot2 x 1 + cot x
−3
√ (x − 1)2 3 x + 4dx
−4
1 − tan(ln x) dx x cos(ln x)
Z 9.
cot(csc(e−x )) dx ex sin(e−x tan(e−x )
Z
9y dy y 3 +1
VI. Consider the region enclosed by the curves y = x3 +
π 2
Z
2x dx √ x (2 + 1) 4x + 2x+1 − 8
1
ey dy e2y − 6ey + 18
Z
ln2 (sin x) dx tan x
10.
8 7y
p
1 − (ln y)2
dy
2 1 1 and x = − y − . A sketch is shown on the right. Set-up 2 2
the integrals representing the following:
1. The perimeter of R2 . 2. The total area of the regions R1 and R2 . 3. The volume of the solid generated when R1 is rotated 1 about the line y = − using the method of washers. 2 4. The volume of the solid generated when R2 is rotated about the line x = −1 using the method of cylindrical shells.
2
VII. Solve the following problems completely. 1. A ball moves along a straight line so that its acceleration (in cm/s2 ) at any time t seconds is given by a(t) = 6t. The ball travels 10 cm to the right of the origin after 2 seconds. Assume that at t = 0, the ball is at the origin. (a) Determine the velocity and position of the ball at any time t. (b) How far has the ball traveled after 4 seconds? What is the speed of the ball at that time? √ √ 2. Find the equation of the line normal to the curve y = tanh sin−1 x − 2 at the point where x = 2. 3. A particle moves along the curve y = ln x so that its abscissa is increasing at a rate of 2 units per second. At what rate is the particle moving away from the origin as it passes through the point (e, 1)? 4. Find the area of the largest rectangle that can be inscribed in the region bounded by the curve y = e−x , the line x = 0 and the positive x-axis.
3