Question 1

Given the function $$y = x^2$ - 2x + 1$$, what is the inverse function $$f^{-1}(x$$)$$ for x$$ \geq 1$$?

Accepted Answer

$$f^{-1}(x) = 1$$ + \sqrt{x + 1}$$

Question 2

Find the domain of the function $$h(x) = -e$$^{-x^2}$$ - 1.$$

Accepted Answer

$$x \in (-\infty, \infty)$$

Question 3

What is the range of the function $$h(x) = -e$$^{-x^2}$$ - 1$$?

Accepted Answer

$$(-2, -1)$$

Question 4

Evaluate $$\lim_{x 	o 0} \sin(x^2$) \cos^2$\left(\frac{1}{x}ight)$$ using the Squeeze Theorem.

Accepted Answer

$$0$$

Question 5

Find the one-sided limit: $$\lim_{x 	o 1^-} \frac{2x - x^2$}{x - 1}.$$

Accepted Answer

$$-1$$

Question 6

Suppose $$f(x) is $$continuous at every $$x$$ and $$\lim_{x 	o 0} \frac{\sin(2x)}{3x} = g. $$For which value of $$c$$ does $$f(x)$$ become continuous at $$x=0 if$$ $$f(0) = c$$?

Accepted Answer

$$c = \frac{2}{3}$$

Question 7

Given the curve $$e^{xy} = (1$$ + \sin $$y)x^2 at $$the point $$(0,0)$$, what is the slope of the tangent line at this point?

Accepted Answer

$$0$$

Question 8

For the function $$y = (2x + g(x))$$, if $$g(1) = 1$$ and $$g'(1) = 2$$, what is $$y'(1)$$?

Accepted Answer

$$4$$

Question 9

Use the linearization of $$f(x) = 	an$$^{-1}$$ x at$$ $$a = 1 to $$approximate $$	an$$^{-1}$$(1.02). $$What is the approximate value?

Accepted Answer

$$\frac{\pi}{4} + \frac{1}{100}$$

Question 10

Let $$f(x) = \ln\left(\frac{x^2$}{x^2$ + 2}\right) on$$ $$[-1, 4]. At $$which $$x$$ does $$f(x)$$ attain its absolute minimum?

Accepted Answer

$$x = 0$$

Question 11

Let $$f(x) = \ln\left(\frac{x^2$}{x^2$ + 2}\right) on$$ $$[-1, 4]. At $$which $$x$$ does $$f(x)$$ attain its absolute maximum?

Accepted Answer

$$x = 4$$

Question 12

If $$f(x) = 10 \sin x - 10x + 10$$, show that $$f(x)$$ has at least one solution in $$(0, \frac{\pi}{2}). $$Which theorem justifies this?

Accepted Answer

Intermediate Value Theorem

Question 13

Given $$f(x) = \ln\left(\frac{x^2$}{x^2$ + 2}\right)$$, what is the critical point in $$[-1, 4]$$?

Accepted Answer

$$x = 0$$

Question 14

Evaluate $$\lim_{x 	o 3} \frac{2 - \sqrt{x - 5}}{x + 3}.$$

Accepted Answer

$$\frac{2}{6}$$

Question 15

If $$f(x) = x^2$ - 2x + 1$$, what is the range of $$f(x)$$ for $$x \geq 1$$?

Accepted Answer

$$[0, \infty)$$

Question 16

Find $$\lim_{x 	o 1^+} \frac{2x - x^2$}{x - 1}.$$

Accepted Answer

$$1$$

Question 17

Given $$f(x) = \ln\left(\frac{x^2$}{x^2$ + 2}\right)$$, what is $$f(4)$$?

Accepted Answer

$$\ln\left(\frac{16}{18}\right)$$

Question 18

If $$f(x) = x^2$ - 2x + 1$$, what is the domain of $$f^{-1}(x$$)$$?

Accepted Answer

$$x \geq 0$$

Question 19

Suppose $$h(x) = -e$$^{-x^2}$$ - 1. $$What is $$\lim_{x 	o \infty} h(x)$$?

Accepted Answer

$$-1$$

Calculus I Midterm Exam Study Guide: Functions, Limits, Differentiation, and Applications

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