sin 2θ = 2 sin θ cos θ

Calculus I Midterm 2 Practice Problems 1. Use the linear approximation of the function f (x) = (x + 1)^1/4 to estimate (1.02)^1/4.

L(x) = 1 + 1

4(x − 0), (1.02)^1/4≈ L(0.02) = 1.005

2. Two hallways, one 8 feet wide and the other 1 feet wide, meet at right angles. Determine the length of the longest ladder that can be carried horizontally from one hallway to the other.

Find the minimum of f (x) = (⁸_x+ 1)²(x²+ 1²)

3. Find the point on the curve y = x² closest to the point (0, 1) Find the minimum of f (x) = (x − 0)²+ (x²− 1)²

4. Suppose a wire 4 ft long is to be cut into two pieces. One will be formed into a square and the other one will be formed into a regular triangle. Find the size of each piece to minimize the total area of the two region.

Find the minimum of A(x) = (4 − x)²+^x₂^√₂³x for x ≥ 0

5. Suppose a 6-ft tall person is 12 ft away from an 18-ft tall lamppost. If the person is moving away from the lamppost at a rate of 2 ft/s, at what rate is the length of the shadow changing?

Sec 3.8: exercise 23.

6. Parametric equations for the position of an object is given. Find the object’s velocity and speed at the given times, and describe its motion.

 x= 2 cos 2t + sin 5t

y= 2 sin 2t + cos 5t (a)t = 0, (b)t = π 2 Sec 3.8: exercise 29

7. Find the derivatice of the function

f(x) = Z cos x

x

p1 − t²dt

Hint: Let g(x) =Rx 0

√1 − t²dt, f (x) = g(cos x) − g(x).

8. If x sin x = Rx²

0 f(t) dt, where f is a continuous function, find f (4). Hint: Differentiate the equation with respect to x.

9. R ₁

x²−4x+3dx

1

2(ln |x − 3| − ln |x − 1|) + C 10. R ₁

x²−4x+5dx

tan⁻¹(x − 2) + C 11. R ln x

x dx

1

2(ln x)²+ C 12. R e^2xsin 2x dx

1

4e^2x(sin 2x − cos 2x) + C

13. R2 0

x² (x²+4)² dx

Hint: Let x = 2 tan θ, change the terms in the integral to sin and cos, and use double-angle formula.

14. R ₁

√x+xdx Hint: Let u =√

x, u²= x.

15. R tan⁴xsec⁴x dx Hint: u = tan θ 16. R tan x dx

=R _{sin x}

cos xdx= − ln | cos x| + C 17. R sec x dx

Hint: multiply with sec x+tan x sec x+tan x

= ln | sec x + tan x| + C 18. R1

−2|2x + 1| dx

=R−1/2

−2 −2x − 1 dx +R1

−1/22x + 1 dx 19. R1

0 x^−1/3dx

= ³₂ 20. R_∞

1 x^−1/3dx DIV

21. R_∞

0 cos x dx DIV 22. R_∞

1

sin x+2 x dx DIV

23. Find the average value of f (x) =√

x on the interval [0, 9]

= ₉₋₀¹ R9 0

√x dx=¹₉²₃x^3/2|⁹0= 2

24. Find the area of the region bounded by y = 2 cos x, y = sin 2x for x ∈ [−π, π]

A=R_−π/2

−π sin 2x − 2 cos x dx +Rπ/2

−π/22 cos x − sin 2x dx +Rπ

π/2sin 2x − 2 cos x dx

25. Let Ω be the region bounded by y = sec x, x = 0, x = ^π₄ and y = 0. Find integrals represent the volume of the solids generated by Ω about (a) x-axis, (b) y-axis, (c) y = −1, (d) x = −1.

(a)Rπ/4

0 π(sec x)²dx (b)Rπ/4

0 2πx sec x dx (c) Rπ/4

0 π[(sec x + 1)²− 1²] dx (d)Rπ/4

0 2π(x + 1) sec x dx

• Double-Angle

sin 2θ = 2 sin θ cos θ

cos 2θ = 2 cos

θ − 1 = 1 − 2 sin

θ

• Derivative formulas

d

dx

sin

x = 1

√ 1 − x

,

d

dx

cos

x = − 1

√ 1 − x

,

d

dx

tan

x = 1

1 + x

,

d

dx

cot

x = − 1 1 − x

,

d

dx

sec

x = 1

|x| √

x

− 1 ,

d

dx

csc

x = − 1

|x| √

x

− 1