What is the requirement for showing work in the Calculus I Midterm?
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You MUST show your work.
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What is the requirement for showing work in the Calculus I Midterm?
You MUST show your work.
What does the Intermediate Value Theorem (IVT) help to prove?
It helps to prove that a function has a root within a certain interval.
What is the condition for a function to be continuous on its domain?
The function must not have any breaks or jumps in its domain.
What is the condition for δ in relation to x for the function f(x) = x/(x - 1)?
If δ < |x - 1|, then |f(x) - 2| < 0.1.
What is the function given in the problem?
f(x) = 4x + 3.
What does it mean for a function to be not differentiable at a point?
It means that the function does not have a defined derivative at that point.
What is the limit of (x^2 - x) as x approaches infinity?
∞.
What is the significance of the limits provided in the midterm question?
They indicate the behavior of the function as it approaches certain values.
What is the definition of the derivative using First Principles?
f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h].
What are the horizontal asymptotes of the function f(x) = (4x^3 - 10x + 7)/(3x^2 + x - 3)?
y = 4/3.
How do you apply the definition of the derivative to f(x) = 4x + 3?
Calculate f(x + h) = 4(x + h) + 3 and then use the limit definition.
What is the limit of (5x^2 - 7)/(2x - 5) as x approaches 0?
-7/5.
What is the derivative f'(x) for the function f(x) = 4x + 3?
f'(x) = 4.
What values of a and b make the piecewise function continuous?
a = 3, b = 2.
Which of the following functions have a domain that includes all real numbers?
f(x) = e^(-x) + x, f(x) = 7sin(x), f(x) = 6/(x^2 - 8), f(x) = 5/(x + 1)
What can be said about the function based on the graph?
There are 2 values for which the function is not differentiable; the function is continuous from the left at x = 0.
What is the first step to find the derivative of the function f(x) = (1 + 2x^2)(3 - 5x^2)?
Use the product rule: f'(x) = u'v + uv', where u = (1 + 2x^2) and v = (3 - 5x^2).
Which of the following limits do NOT exist?
lim (x → 0) (x^3 - 7)/(x^2 - 4); lim (x → ∞) (x^2 - 10)/(x^2 + 2)
How do you find the derivative of the function f(x) = x^2 * e^(3x)?
Apply the product rule: f'(x) = u'v + uv', where u = x^2 and v = e^(3x).
What does the Intermediate Value Theorem guarantee for a continuous function on the interval [1, 5]?
There exists at least one c in [1, 5] such that f(c) = 2.
What is the limit of (1 - x)/(x - 1) as x approaches 1?
The limit is undefined.
What is the formula to find the derivative of a composite function f(x) = g(h(x))?
Use the chain rule: f'(x) = g'(h(x)) * h'(x).
Which statements about the graphs of f(x) and g(x) are true?
g(x) is the derivative of f(x); g(x) is differentiable everywhere; g(x) is an odd function; f(x) is an even function.
What are the three properties that define continuity of a function f(x) at x = a?
Given g(10) = 4 and h(10) = 560, how do you find f'(10) if g'(10) = 0 and h'(10) = 35?
Use the chain rule: f'(10) = g'(h(10)) * h'(10) = 0 * 35 = 0.
What does the Squeeze Theorem help to determine?
It helps to find the limit of a function by 'squeezing' it between two other functions.
What does lim (x → ∞) f(x) indicate?
It indicates the behavior of the function f(x) as x approaches infinity.
What does lim (x → 6) f(x) indicate?
It indicates the behavior of the function f(x) as x approaches 6.
What does lim (x → 1) f(x) indicate?
It indicates the behavior of the function f(x) as x approaches 1.
How can you determine the number of discontinuities in a function f?
By analyzing the limits and the defined points of the function.
