Math 140 Practice problems for Test 2
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Math 140
Practice problems for Test 2
The exam will contain a number of problems, of types similar to the problems given in this review sheet (but not that many). Please prepare for each problem type.
1. Find the derivative of the given function.
(a) ln(x3 +1)
(b) ln(x^x +1)
(c) 3x8 +x2 2．+ln(5)+4 +secx +x 
(d) sin(2x)cos(3x)
(e) etant
(f) f (x) = sec(x2 )
(g) s(t) = arctan(t3 )
(h) y = arcsin(^x)
(i) x2 arcsec(x)
2. Let x2 +2xy +42 =y 3 define y as an implicit function of x. (a) Find dy/dx using implicit differentiation.
(b) Find the points on the curve x2 +2xy +4y2 = 3 where the tangent line is horizontal.
3. Verify that the coordinates of the given point satisfy the given equation. Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
(a) y sin(2x) = x cos(2y), ╱ 2(几) ﹐ 4(几) 、
(b) x2 +xy +y2 = 3, (1﹐ −2)
(c) x2 +2xy − y2 + x = 2, (1﹐ 2)
4. (a) Find the linearization of f (x) =^x at a = 100 and use it to approximate ^99．8. (b) Find the linearization of f (x) =^8 +x at a = 1 and use it to approximate ^9．02.
(c) Find the linearization of f (x) =^38 +x at a = 0 and use it to approximate ^37．97.
(d) Find the linearization of f (x) = lnx at a = 1 and use it to approximate ln 1．01.
(e) Use a linear approximation to estimate (0．9999)2022 .
5. The pH p of a solution where the concentration of hydronium ions is C is given by
p = −log10C．
How quickly is the pH changing when the pH is 7 and the concentration is increasing by 3 × 10−8 per second?
6. Let f (x) = x + 
, where K is a constant. Find the absolute minimum and maximum of f (x) on the interval
[1﹐ K] assuming that K ≥ 2. (Your answers will be in terms of K.)
7. A sugar cube slowly dissolves while keeping its cubic shape. The volume is decreasing by 0．03cm3 per second. How quickly is the height of the cube decreasing when the cube has volume 
cm3 ? Include appropriate units.
8. An object of mass m kg that is moving at velocity v meters per second has kinetic energy 
mv2 Joules. Suppose that a 4 kg mass has positive velocity and is accelerating, and at the moment when the kinetic energy is 200 Joules, the kinetic energy is increasing at a rate of 120 Joules per second. Find the acceleration at this moment in time. Include appropriate units.
9. Suppose that f (x) is continuous and differentiable everywhere, and that f \ (x) ≤ 3 for all x. Is it possible that f (2) = 10 and f (6) = 25? Explain.
10. Recall that given a function f satisfying certain conditions, the Mean Value Theorem relates the slope of a secant line of the graph of f to the slope of a tangent line to f at some intermediate point(s) c.
Give the precise statement of the Mean Value Theorem.
For the given functions and the given intervals, find the point(s) c that satisfies the conclusion of the Mean
Value Theorem.
(a) f (x) = x2 , for the interval [1﹐ 3].
(b) f (x) =^x, for the interval [4﹐ 9].
(c) f (x) = lnx, for the interval [1﹐ e].
(d) f (x) = sinx, for the interval [0﹐ 几].
11. Find the critical points of the function.
(a) f (x) = x
1 +x2 .
(b) f (x) = 2x3 − x2 − 20x + 1.
(c) f (x) = x2 e −x .
12. Find the maximum and minimum values of f on the given interval.
(a) f (x) = 2x3 − x2 − 20x +1, x ∈ [−4 ﹐ 3].
(b) f (x) = x2 e −x , x ∈ [1﹐ 3].
2023-04-05