Farber Practice for Midterm test Math 141

The questions on the midterm correspond to questions from Test 1 and Test 2. Answers with explanations follow.

1)Find the slope of a line tangent to the curve of where

2) y =

3) y =

4) y = ln

5) y = (x sin x)

6) y = (ln 4x)

7) y = 2 ( 4)

8)y = ln (sinh 5x)

9) y = ln x -

10) y = 5 x

11) y =

12) y = 2x - 2

13) y =

14) y = cosh

39) f(x) = ; f(x).

40)

41)

42)

43)

Answers with explanations.
1) 0.973. Recall that the slope of the tangent is equal to .
2) . Take the natural log on both sides. Then differentiate with implicit differentiation. Solve for .
3) .

Write in terms of the natural log and use rules of logs to simplify before differentiating.
4) . Use rules of logs to simplify into the difference between two logs. Then differentiate
5) . Convert to natural log. Remember than ln5 is a constant, so factor out . Use the chain rule with the product rule. (x sin x)==ln(x sin x). There are several equivalent ways to write the answer.
6) . Use the chain rule.
7) . Use the chain rule.
8) 5 coth 5x. use the chain rule.
9)  - + 5ln x
10)   + 15 x. Use the product rule.
11)  (ln cos x - x tan x). Take the natural log on both sides. Then simplify using the laws of logs. The right side will be . Solve for and write in terms of x,
12) 2x
13) . Use the chain rule.
14) 4 sinh . Use the chain rule.
15) 4 ln + + C. Use partial fraction decomposition. Remember you have a repeated linear factor here so you must account for denominators for the first and second power of this factor.
16) (3z2 - 7)3/2 + C This is a simple Calc I u-substitution. u = 3 and du = 6z dz.
17)  - . Let u= x. So du =
18) x + sin 2x + C. Use the power reducing trig identity. 2x = 1+ cos 2x. The latter is easy to integrate. .
19)  tanh ( 6x - 5) + C. Let u = 6x-5. Then du = 6dx.
20) tan2 x + C. let u=tan x and du = x dx or change to sines and cosines.
21) x - sin 2x + C. Use the power reducing formula to rewrite 2 x = 1- cos2x
22)  ln + C. Use trig substitution with u = 3 sin θ and du = 3 cos θ. After you integrate the trig function, back substitute using a right triangle relationship from the original substitution.
23) 5 sinh + C. Let u = . So du = 5dx
24) ln 3x - + C. Integration by parts. u = ln3x and dv=dx
25) -x+ +C. Break off a factor of cos x. Use the Pythagorean identity to rewrite x. You will have two terms to integrate. Both contain a power of sin x multiplied by cos x. Use u = sin x and du = cos x dx.
26)   + ln + C. Split the numerator into two fractions. Each can be integrated separately.
27)  + C. Recall that ln 3 is a constant. Move it to the left of the integral sign.
28)   + + C. Use trig substitution. a=7 and u=x and x=7sinθ and dx= 7 cosθ.
29) ln + C. Use partial fractions. You can use laws of logs to simplify.
30) - + C. Let u = . Then du = -dx. The constant can be moved through the integral sign.
31) (1/2)x2e2x - (1/2)xe2x + (1/4)e2x + C. Use integration by parts with u = and dv = . From this result, you will have to use integration by parts again.
32) - 2 + 3 + C. Split the numerator so you have two fractions. To integrate the first use a simple u-substitution of u = 36 - and du = -2x dx. For the second, use a trig substitution where x = 6 sin θ and dx = 6 cos θ dθ.
33)   + C. Let a = 36, u = 25x. So du = 25dx
34) 6 ln + C. Use trig substitution with x = sec θ dx = sec θ tan θ. Draw a right triangle to help by back substitute.
35) 3 ln + ln + + C. Use Partial Fraction decomposition. The denominator factors into x(+25). So the equivalent to the integrand is + . Find A, B and C and then integrate.
36)  + C. Rewrite in terms of natural log. Then let u = x+5 and du=dx.
37) - + C. Use trig substitution with x = 4tan θ and dx =4 θ d θ. To evaluate the resulting trig integral, put everything in terms of sin θ and cos θ and use u- substitution. Form a right triangle to back substitute so final answer is a function of x.
38) . Rewrite integrand as . If u = the du =3dx. Match this with the inverse tangent formula.
39) 1. Use l'hopital's rule.
40) let y = and take logarithms of both sides ln y = ln x =

ln y = . = differentiate

= -2 = 0 So = = = 1
41)1. Write as y = .Take the natural log on both sides. On the right side use the laws of logs to rewrite and eliminate the exponent. Rewrite so it is in a or type form so you can use L'Hopital's rule. Evaluate the limit on the right. That is equal to lny. Solve for y by taking e to the power on both sides.
42)Using l'Hopital's rule

=

= = = -1
43)- . Use L'hopital's rule twice.

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