Q: Evaluate the following limits and identify the horizontal asymptotes (if any) for $$f(x) = \frac{5x}{25x + 3}$$: $$\lim_{x \to \infty} f(x)$$ and $$\lim_{x \to -\infty} f(x).$$

$$\lim_{x \to \infty} f(x) = \frac{1}{5}$$, $$\lim_{x \to -\infty} f(x) = \frac{1}{5}$$; Horizontal asymptote: $$y = \frac{1}{5}$$

Q: Find the derivative of $$f(x) = \cos(\tan$$^{-1}$$(10 \ln x)).$$

$$\frac{-10 \sin(\tan$$^{-1}$$(10 \ln x))}{x(1 + (10 \ln x)^2$)}$$

Question 1

Given the function $$r(U) = \frac{1}{5} \left( \sqrt{225 + \frac{5U}{\pi}} - 15 ight)$$, calculate $$\lim_{U 	o 0^+} r(U)$$ and provide an interpretation in the context of a right cylinder with height 15 cm and surface area $$U cm$^2. $$What does this limit represent?

Accepted Answer

$$\frac{1}{5} \sqrt{210}$$; As $$U$$ approaches 0 from the positive side, the radius of the cylinder approaches $$\frac{1}{5} \sqrt{210} cm.$$

Question 2

Refer to the graph of $$f(x) ($$first image). What is $$\lim_{x 	o 5} f(x)$$?

Accepted Answer

$$5$$

Question 3

On the interval $$(0, 15)$$, locate the points where the function $$f(x) ($$second image) has discontinuities. For each discontinuity, indicate which continuity conditions are not met.

Accepted Answer

At $$a = 3$$, the function is discontinuous because $$\lim_{x 	o a} f(x) 
eq f(a)$$; at $$a = 10$$, the function is discontinuous because the limit does not exist.

Question 4

Let $$g(x) = \begin{cases} \frac{x^3$ - 1}{x - 1} & x 
eq 1 \ a & x = 1 \end{cases}. $$For what value of $$a is$$ $$g(x)$$ continuous at $$x = 1$$?

Accepted Answer

$$3$$

Question 5

Evaluate the following limits and identify the horizontal asymptotes (if any) for $$f(x) = \frac{5x}{25x + 3}$$:
$$\lim_{x 	o \infty} f(x)$$ and $$\lim_{x 	o -\infty} f(x).$$

Accepted Answer

$$\lim_{x 	o \infty} f(x) = \frac{1}{5}$$, $$\lim_{x 	o -\infty} f(x) = \frac{1}{5}$$; Horizontal asymptote: $$y = \frac{1}{5}$$

Question 6

Use the following theorem to evaluate $$\lim_{x 	o 0} \frac{\sin 39x}{9x}$$:
Given $$\lim_{x 	o 0} \frac{\sin x}{x} = 1.$$

Accepted Answer

$$\frac{13}{3}$$

Question 7

Find the derivative of $$f(x) = \cos(	an$$^{-1}$$(10 \ln x)).$$

Accepted Answer

$$\frac{-10 \sin(	an$$^{-1}$$(10 \ln x))}{x(1 + (10 \ln x)^2$)}$$

Question 8

Find the value of $$h'(e)$$ where $$h(x) = (3^x)$$ and $$e is $$the base of the natural logarithm.

Accepted Answer

$$(3^e) \cdot \frac{1}{3e \ln 3} \cdot 3e^2$

Question 9

Determine the equation of the line tangent to the graph of $$y = x \sqrt{17 - x^2$} at $$the point $$(1, 4).$$

Accepted Answer

$$y = \frac{15}{4} x - \frac{1}{4}$$

Question 10

Consider an object whose position is given by $$p(t) = 5t^2$ - 10t$$, where $$p is in $$meters and $$t is in $$seconds. Find the velocity and acceleration of the object at $$t = 4.$$

Accepted Answer

Velocity: $$30 m/s$$, Acceleration: $$10 m/s$^2$$

Question 11

Let $$f(x) = h(g(x)). $$Calculate $$f'(4)$$ using the following table (not shown, but assume $$h'(g(4)) = 5$$ and $$g'(4) = 4).$$

Accepted Answer

$$20$$

Question 12

Find the derivative of the function $$f(x) = \left(3 + \frac{1}{x}\$$right)^{2x}$$ using logarithmic differentiation.

Accepted Answer

$$f'(x) = 2 \left(3 + \frac{1}{x}\$$right)^{2x}$$ \ln\left(3 + \frac{1}{x}\right) + 2 \left(3 + \frac{1}{x}\$$right)^{2x - 2}$$

Question 13

Use the following limit definition to determine the slope of the line tangent to the graph of $$f(x) = \frac{19}{x^3$} at$$ $$P(-1, -19)$$:
$$\displaystyle m_{	ext{tan}} = \lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$$

Accepted Answer

$$-38$$

Question 14

A pebble is dropped from a bridge, and the distance it falls is given by $$d(t) = 15t^2$, where d$ is in meters and t$ is in seconds. If the pebble takes 7$ seconds to reach the water below, what is the height of the bridge? Also, calculate the pebble's speed upon hitting the water in kilometers per hour.

Accepted Answer

$$735 m$$, $$189 km/h$$

Question 15

A basketball is shot upwards from the ground with an initial velocity of $$40 ft/s. $$The height (in feet) of the basketball $$t$$ seconds after it is shot is given by $$h(t) = -16t^2$ + 40t. $$What is the maximum height reached by the basketball?

Accepted Answer

$$35 ft$$

Question 16

The total surface area of a cylinder having radius $$r$$ and height $$h is $$given as $$A = 2\pi r(r + h). $$Calculate the value of $$\frac{dr}{dh} if$$ $$A = 500\pi$$, $$r = 5$$, and $$h = 12.$$

Accepted Answer

$$-\frac{5}{22}$$

Question 17

Verify the following equation using differentiation:
$$\frac{d}{dx}(x^5$ - \ln(x^3$ + 2)) = 5x^4$ - \frac{3x^2$}{x^3$ + 2}$$

Accepted Answer

True

Question 18

Determine the derivative of the function $$f(x) = 4x^4$ using the definition:
$$\displaystyle f'(x) = \lim_{h 	o 0} \frac{f(x + h) - f(x)}{h}$$

Accepted Answer

$$16x^3$$

Question 19

Determine the linear approximation of $$h(y) = \sqrt{4y - 5} at $$the point $$a = 2.$$

Accepted Answer

$$L(y) = \frac{2\sqrt{3}}{3} y - \frac{\sqrt{3}}{3}$$

Question 20

Consider the function $$f(x) = x \sqrt{9 - x^2$} on $$the interval $$[-3, 3]. $$Find the critical points of $$f$$ and use the First Derivative Test to classify these points. Then, determine the absolute maximum and minimum values of $$f on $$the specified interval (if there are any).

Accepted Answer

Critical points: $$x = \pm \frac{3\sqrt{2}}{2}$$; Local maximum at $$x = \frac{3\sqrt{2}}{2}$$ and local minimum at $$x = -\frac{3\sqrt{2}}{2}$$; Absolute maximum: $$4 at$$ $$x = \frac{3\sqrt{2}}{2}$$; Absolute minimum: $$-4 at$$ $$x = -\frac{3\sqrt{2}}{2}$$

Calculus Final Exam Study Guide: Limits, Continuity, and Derivatives

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