Math 152-02 and 04: Sample Test Problems for Midterm I
1. Find a formula for the inverse of the function f(x) = 1+2x
3−x.
2. Find the tangent line equation of f−1(x) at x=f(π/2) = π, where f(x) =
2x+ cos x.
3. Solve the equation ln(x−1) + ln(x+ 1) = 1 for x.
4. Evaluate the limits:
a. lim
x→1
x−1
tan(x−1), b. lim
x→1(x−1) cot(x−1), c. lim
x→∞
exp(x)
x2,
d. lim
x→0
1−cos x
x2, e. lim
x→0+√xln(x), f. lim
x→∞
x(1/x)
5. Find the derivative of the following functions
a. f(x) = pln(x), b. g(x) = arcsin( 1
x), c. h(x) = exp(cos x) + sin(exp(−x))
d. k(x) = tan−1(2x), e. y =x√2x, f. m(x) = ln(ln(x))
6. Find the area of the region bounded by the curves = 2x, y = 5x, x =−1, x =
1.
7. Find an equation of the tangent line to the curve y=xln(2x) at (e, e).
8. Find the volume of the solid obtained by rotating the region under the curve
y=1
1+x2from 0 to 3 about the y-axis.
9. Evaluate the following integrals
a. Zx+ 2
x2+ 4dx, b. Zdx
√2−2x−x2, c. Ze−xsin(e−x)dx, d. Z(ln(x))2
xdx
10. Evaluate the integral R1
0x2exdx (Hint:Use Integration By Parts)
11. Evaluate the following trigonometric integrals
a. Zsin3xcos2xdx, b. Ztan3xsec xdx, c. Zπ
0
cos2(2x)dx
12. Evaluate the following integrals using trigonometric substitution
a. Z√1 + x2
xdx, b. Z3/2
0
1
√9−4x2dx
13. Express the following integrands as a sum of partial fractions and then
evaluate the integrals
a. Zx+ 1
x(x−1)dx, b. Z4x2+x+ 2
x3+ 4xdx
14. Evaluate the following improper integrals
a. Z0
−∞
xexdx, b. Z3
0
1
t−1dt
1