Question 1

Evaluate the following limit using algebraic simplification:

$$\lim_{h 	o 0} \frac{\sqrt{9 + h} - 3}{h}$$

Accepted Answer

$$\frac{1}{6}$$

Question 2

Let $$f(x) = \begin{cases} x^2$ + 1, & x \u003c -10 \ -10x + 10, & x \geq -10 \end{cases}.

$$Find $$\lim_{x 	o -10^-} f(x).$$

Accepted Answer

$$101$$

Question 3

Find the derivative of $$h(x) = 4x^2$ + 3x^2$ - 5.$$

Accepted Answer

$$14x$$

Question 4

Suppose $$s(t) = t^3$ - 9t^2$ + 24t$$ measures the position of an object for $$0 \leq t \leq 6. $$When is the object stationary?

Accepted Answer

$$t = 3$$ and $$t = 4$$

Question 5

Use implicit differentiation to find $$\frac{dy}{dx}$$ for $$5x^4$$ + $$3y^2 = 9xy$.

Accepted Answer

$$\frac{20x^3$ - 9y}{6y - 9x}$$

Question 6

A spherical balloon is inflating with helium at a rate of $$800\pi ft$^3/$$min. How fast is the radius increasing when the radius is 5 ft? (Volume of a sphere: $$V = \frac{4}{3}\pi r^3$)

Accepted Answer

$$\frac{8}{5} ft/$$min

Question 7

Let $$f(x) = 2x^3$ + 3x^2$ - 120x + 3. $$Find all critical points.

Accepted Answer

$$x = -10, x = 2$$

Question 8

For $$f(x) = 3x^4$ - 2x^3$ + 8$$, determine the interval(s) where $$f(x) is $$concave up.

Accepted Answer

$$(-\infty, \frac{1}{2}) \cup (1, \infty)$$

Question 9

Evaluate the limit using L'Hôpital's Rule:

$$\lim_{x 	o 6^-} \frac{x^3$ + 5x^2$ + 6x}{x^2$ - 36}$$

Accepted Answer

$$\infty$$

Question 10

Find $$f(x) if$$ $$f'(x) = 12x - 5$$ and $$f(0) = 5.$$

Accepted Answer

$$6x^2 - 5x + 5$

Question 11

Evaluate the definite integral:

$$\$$int_0$$^1 x^2$ - 3x + 7\ dx$$

Accepted Answer

$$\frac{19}{3}$$

Question 12

Evaluate $$\frac{d}{dx} \$$int_0$$^{x^2$} (t^2$ + 13)\ dt.$$

Accepted Answer

$$2x(x^2 + 13$$)$$

Question 13

Find the area of the region between $$y = x^2$ - 19$$ and $$y = 18x.$$

Accepted Answer

$$\frac{6859}{12} ($$after evaluating the definite integral)

Question 14

Find the volume of the solid generated by revolving the region between $$y = 49 - x^2$ and y = 0$ around the x-axis.

Accepted Answer

$$\frac{7688}{3}\pi ($$using the disk method)

Question 15

Let $$R be $$the region bounded by $$y = 4x - x^2$ and y = 0$. Use the shell method to find the volume of the solid generated when R$ is rotated around the y-axis.

Accepted Answer

$$32\pi$$

Question 16

Evaluate the indefinite integral: $$\int \sin(7x + 2)\ dx.$$

Accepted Answer

$$-\frac{1}{7}\cos(7x + 2) + C$$

Question 17

Evaluate the indefinite integral: $$\int \frac{x^4$}{3x^5$ + 11} dx.$$

Accepted Answer

$$\frac{1}{15} \ln|3x^5$ + 11| + C$$

Question 18

A car's velocity is given by $$v(t) = 3t^2$ - 18t + 24. $$For $$0 \leq t \leq 6$$, when is the car moving to the left?

Accepted Answer

$$3 \u003c t \u003c 4$$

Question 19

Let $$f(x) = 3x^4$ - 2x^3$ + 8. At $$which $$x-$$value(s) does $$f(x)$$ have an inflection point?

Accepted Answer

$$x = \frac{1}{2}$$ and $$x = 1$$

Math 140 (Calculus I) Final Exam Study Guidance

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