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