Question 1

Evaluate the following limit:

$$\displaystyle \lim_{x 	o 0} \frac{\sin 3x}{\sin 5x}$$

Accepted Answer

$$\displaystyle \frac{3}{5}$$

Question 2

Evaluate the following limit:

$$\displaystyle \lim_{x 	o 0} \frac{e$$^{3x}$$ - 1}{x(x^2$ - 4)}$$

Accepted Answer

$$-\frac{3}{8}$$

Question 3

Which of the following correctly states the epsilon-delta definition of $$\displaystyle \lim_{x 	o 0} \sin(5x) = 0$$?

Accepted Answer

For every $$\varepsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that if $$0 \u003c |x| \u003c \delta$$, then $$|\sin(5x)| \u003c \varepsilon.$$

Question 4

Suppose $$f(x) = e^x - 4 + x^3$. Use the Intermediate Value Theorem to determine which statement is true about f(x$$)$$ on $$[0,2]$$.

Accepted Answer

$$f(x)$$ changes sign on $$[0,2]$$, so $$e^x = 4 - x^3$ has a solution in $$[0,2]$$.

Question 5

What is the relationship between the slope of the tangent line to a curve $$f at a $$point $$a$$ and the derivative $$f'(a)$$, assuming $$f is $$differentiable at $$a$$?

Accepted Answer

The slope of the tangent line at $$x=a is $$equal to $$f'(a).$$

Question 6

Find the equation of the tangent line to the curve $$f(x) = 3x^2$ - 2x + 1 at$$ $$x = -1.$$

Accepted Answer

$$y = -8x - 9$$

Question 7

Given $$x^2$$ + $$y^2 - 3y = 0$ where x$ and y$ are differentiable functions of t$, use implicit differentiation to find $$\displaystyle \frac{dy}{dx}$$.

Accepted Answer

$$\displaystyle \frac{-2x}{2y - 3}$$

Question 8

Using the Bisection Method, which interval would you start with to approximate $$2^{1/3}$$ with error less than $$\frac{1}{8}$$?

Accepted Answer

$$[1, 2]$$

Question 9

Suppose you are using the Bisection Method to approximate a root of $$f(x) = x^3$ - 2. $$After three iterations starting with $$[1,2]$$, what is the maximum possible error?

Accepted Answer

$$\displaystyle \frac{1}{8}$$

Question 10

Which of the following is a correct application of the Intermediate Value Theorem?

Accepted Answer

If $$f is $$continuous on $$[a,b]$$ and $$f(a) \u003c 0 \u003c f(b)$$, then there exists $$c \in (a,b)$$ such that $$f(c) = 0.$$

Question 11

A medical researcher models the spread of a virus with $$N(t) = e$$^{kt}$$, where N$ is the number of cases at time t$ and k$$ \u003e 0$$. What does the derivative N$$'(t)$$ represent?

Accepted Answer

The instantaneous rate of change of cases at time $$t.$$

Question 12

Suppose $$y is $$implicitly defined by $$x^2$$ + $$y^2 - 3y = 0$. If x = 1$ and y = 2$, what is $$\displaystyle \frac{dy}{dx}$$ at that point?

Accepted Answer

$$-1$$

Question 13

Which of the following best describes the tangent line to $$f(x) at$$ $$x=a$$?

Accepted Answer

It is the line that best approximates $$f(x)$$ near $$x=a.$$

Question 14

A chemical reaction is modeled by $$C(t) = 3t^2$ - 2t + 1$$, where $$C is $$the concentration at time $$t. $$What is the rate of change of concentration at $$t=2$$?

Accepted Answer

$$10$$

Question 15

If $$f(x) = 3x^2$ - 2x + 1$$, what is the $$y-$$intercept of the tangent line to $$f at$$ $$x = -1$$?

Accepted Answer

$$-9$$

Question 16

Which of the following is NOT a requirement for applying the Intermediate Value Theorem to a function $$f on$$ $$[a,b]$$?

Accepted Answer

$$f$$ must be differentiable on $$[a,b].$$

Calculus I Midterm Study Guidance: Limits, Derivatives, Tangents, Implicit Differentiation, and Numerical Methods

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