Limits and Continuity
What is/are the vertical asymptotes of 1/(x2 - 25)?
x = -5, x = 5
If y = 2x - x2 + x100, then what is dy/dx?
dy/dx = 2 - 2x + 100x99
If f(x) = e-2x, then what is f'(0)?
-2
If y = x2 - 4x, then what is/are the critical numbers?
x = 2
What is the antiderivative of y = e5x?
e5x/5
If f(x) = |x - 1| / (x - 1), then what is the limit as x approaches 1?
DNE
If f(x) = (1 + x)1/2, then what is f'(1)?
1/(2√2)
OR
√2/4
If y = log7(x) + 𝜋, then what is dy/dx?
dy/dx = 1/(xln(7))
On what interval(s) is y = e-x decreasing?
(-∞, ∞)
If f(x) = sec2(x) - sin(x), then what is the F(x), the antiderivative?
F(x) = tan(x) + cos(x) + C
For y = cos(1/x), what kind of discontinuity occurs at x = 0?
Oscillating discontinuity
If y = cos(x) + sin(x) + x/2, then what is dy/dx at x = 𝜋/6?
√3/2
If f(x) = tan-1(x), then what is the slope of the tangent line at x =√2?
1/3
The volume of a sphere is changing with respect to time, what is dV/dt?
dV/dt = 4𝜋r2(dr/dt)
What is the area bounded by the curve y = 3𝜋x2 and the x-axis on [0, 1]?
𝜋
Explain why the Intermediate Value Theorem cannot be used to prove the existence of a root for f(x) = 1/x on [-1, 1].
f(x) is not continuous at x = 0
If y = x1/2, then what is f'(0)?
Not differentiable at x = 0
1
If f(x) = ln(x) / (x - 1), then what is f'(1)?
1
If y = 1/x, then what is the area bounded by the curve and the x-axis on [1, e2020]?
2020
If y = tan(x), then what is the limit as x approaches 𝜋/2 from the right?
-∞
If y = cos(x2) + 5x + sin(x), then what is dy/dx at x = 0?
dy/dx = 6
If y = 10x + ln(x) + cos-1(x), then what is dy/dx?
dy/dx = 10xln(10) + (1/x) - 1/(1 - x2)1/2
A particle moves along the x-axis so that for t ≥ 0, its position (in metres) after t seconds can be modelled by the function x(t) = x2 - cos(x) + 𝜋. What is the acceleration after t = 𝜋 seconds?
1 m/s2
If y = 2/√x, then what is the volume of the solid that results from rotating about the x-axis the area bounded by the curve and x-axis on [1, e]?
4𝜋