Calculus Practice: Integrals, Limits, Derivatives, and Applications

What is the result of ∫[1 to 5] (1 + √(4 − (x−3)^2)) dx?

4 + 2π

What is the result of ∫[0 to √3] (1 + √(4 − x^2)) dx?

(3√3)/2 + (2π)/3

What is lim n→∞ Σ[k=1..n] k/(n^2 + k^2)?

ln2/2

What is lim n→∞ Σ[k=1..n] n/(n^2 + k^2)?

π/4

What is F′(1/2) if F(x)=∫[x to x^2] cos(πt) dt?

√2/2

What is d/dx [ ∫[e^4 to e^(2x)] (√x)(ln t) dt ]?

4xe^(2x) - 8e^ 4√x

What is the result of ∫[0 to π/4] (1 + e^(tan x)) sec^2(x) dx?

e

What is the result of ∫[0 to 1] (arctan x)/(1+x^2) dx?

π^2/32

If 80% decays in 5 hours, what is the half-life h (hours)?

h = 5 ln(1/2)/ln(0.20)

If Carbon-14 has 73% remaining and a half-life of 5715 years, how do you find age t?

t = 5715·ln(0.73)/ln(1/2)

What are the solutions to 2sin^2(x) = cos(2x) on [0,2π]?

x = π/6, 5π/6, 7π/6, 11π/6

What are the solutions to sin(2x) − cos(x) = 0 on [0,2π]?

x = π/6, 5π/6, π/2, 3π/2

What is y′ if y=(x^(3/4))^(tan x)?

y′= ¾ (x³/4)^tanx (sec²xlnx+tanx/2)

What is y′ if y = x^(ln(2x^3))?

y′ = x^(ln(2x^3))·((3lnx+ln(2x³))/x)

What is the rate at which the shadow length is shortening if a 6 ft person walks toward a 24 ft streetlight at 3 ft/s?

-1 ft/s

If a sand pile has dV/dt=12π ft^3/min and is a cone with diameter=height, what is dh/dt when h=6?

dh/dt = 4/3 ft/min

If a conical cup has r=8, h=16 and dV/dt=8 cm^3/s, what is dh/dt when h=8?

dh/dt = 1/(2π) cm/s

What is dθ/dt when a 15-ft ladder's bottom is pushed toward a wall at 3 ft/s and θ=π/4?

dθ/dt = √2/5 rad/s

What is the linear approximation of ln(1+x) at x=0 for x=0.01?

0.01

What is the linear approximation of e^(2x) at x=0 to estimate e^(1.01)?

2.01

What is the maximum area of a right triangle with a hypotenuse of 7?

49/4

What is the best distance from a screen if the bottom is 5 ft above the eye and the top is 20 ft above the eye?

10 ft

What is the maximum area of triangle ABC with bars AO=BO=1 and angle θ?

(3√3)/8

What is the maximum area of a rectangle under y = 4/(4+x^2) with base on the x-axis?

2

Is f(x)= {5x if x<1; x^3+4 if x≥1} continuous and differentiable at x=1?

Continuous: Yes. Differentiable: No

Is h(x)=|x−2|(x−5)|x−5| continuous and differentiable at x=2 and x=5?

x=2: continuous Yes, differentiable No. x=5: continuous Yes, differentiable Yes

What values of a and b make f(x)= {2x − a if x≤0; (√(x^2+4) − b)/(x^3+x^2) if x>0} continuous?

a = −1/4, b = 2

What values of a and b make f(x)= {x − b + 1 if x≤4; (ax − b)/(x−4) if x>4} continuous?

a = 1, b = 4

What is f^(-1)(x) and the range of f^(-1) for f(x)=1−2x on (6,∞)?

f^(-1)(x)=(1+6x)/(x+2) ; Range of f^(-1): (6,∞) ; Domain is (-2,∞)

What is f^(-1)(x) and the range of f^(-1) for f(x)= e^x/(2e^x+1)?

f^(-1)(x)=ln(x/(1−2x)) ; Range of f^(-1): (−∞,∞)

What is dy/dx for e^(xy) = 7x − y?

dy/dx = (7 − y e^(xy)) / (1 + x e^(xy))

What is dy/dx for ln(x/y) = π(7x − y)?

dy/dx = y(7π − 1/x)/(πy − 1)

What is f′(π/6) if g′(1/2)=5 and f(x)=g(2sin^2 x)?

5√3

What is F′(4) if F(x)=1/g(f^(-1)(x)) with f(2)=4, f′(2)=8, g(2)=3, g′(2)=2?

−1/36

What is the velocity when height is 12 ft on the way up for h(t)=32t−16t^2?

16 ft/s

What is the speed when the rocket hits the ground for y(t)=20(t−t^2)e^t?

20e m/s

What are the local mins and maxes for f′(x)=(x+2)^2(x+3)^3(x−1)^2(x−2)(x−4)^5?

Local mins at x=−3,4 ; Local max at x=2

What are the local extrema and inflection points for f′(x)=(x−2)^3(x+2)^3?

Local max x=−2 ; Local min x=2 ; Inflection x=0

What are the key features for the graph of y = x/(1−x^2)?

VAs: x=±1 ; HA: y=0 ; intercept: (0,0)

What are the key features for the graph of y = x/(4x^2−25)?

VAs: x=±5/2 ; HA: y=0 ; intercept: (0,0)

What are the absolute min and max for f(x)=4cos^3x+21cos^2x−24cosx on [−π/2, π/2]?

Abs min = −25/4 ; Abs max = 1

What are the absolute min and max for f(x)=x−3ln(x) on [1,4]?

Abs min = 4−3ln4 ; Abs max = 1

How many c satisfy the Mean Value Theorem for f(x)=x^4−2x^2+5x+3 on [−2,2]?

3

What is f′(x) for f(x)=arctan(x) − arctan(−1/x)?

f′(x)=0 (x≠0) ; f(7)=π/2 ; f(−7)=−π/2

What is lim x→∞ (x^2 − √(x^4+9x^2))?

−9/2

What is lim x→0 ln(cos(2x))/x^2?

−2

What is lim x→0 (2x − tan(2x))/(4x − sin(4x))?

−1/4

What is lim x→∞ [ x·ln( (2 arctan x)/π ) ]?

−2/π

What is lim x→0 (1 + 3sin x)^(7/x)?

e^21

What is lim x→4+ (x/4)^(12/(x−4)) and lim x→3+ (x/3)^(6/(x−3))?

e^3