Q: Find the equation of the tangent line to $$y = \ln(x^2$ + 1) at$$ $$x = 1.$$

$$y = (x - 1) + \ln(2)$$,\text{ where the slope is }1$

Question 1

Which of the following statements best describes the conditions under which L'Hôpital's Rule can be applied to evaluate $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$?

Accepted Answer

Both $$f(x)$$ and $$g(x)$$ must be differentiable near $$x = a$$, $$g'(x) 
eq 0$$ near $$x = a$$, and $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$ must be of the form $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}.$$

Question 2

Evaluate the following limit using L'Hôpital's Rule: $$\lim_{x 	o 0} \frac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 3

A function $$f(x) is $$defined as follows:

$$\displaystyle f(x) = \begin{cases} x^2$ + 2x & x \u003c 1 \ a x + b & x \geq 1 \end{cases}$$

For what values of $$a$$ and $$b is$$ $$f(x)$$ continuous for all $$x$$?

Accepted Answer

$$a = 4,\ b = -3$$

Question 4

Suppose the position of a particle is given by $$s(t) = t^3$ - 6t^2$ + 9t + 2. $$What is the velocity and acceleration of the particle at $$t = 2$$?

Accepted Answer

Velocity: $$-3$$, Acceleration: $$-6$$

Question 5

Find the equation of the tangent line to $$y = \ln(x^2$ + 1) at$$ $$x = 1.$$

Accepted Answer

$$y = (x - 1) + \ln(2)$$,	ext{ where the slope is }1$

Question 6

A 50-ft ladder is placed against a building. The base of the ladder slides away from the building at 3 ft/min. How fast is the top of the ladder sliding down the wall when the base is 30 ft from the building?

Accepted Answer

$$-2 ft/$$min

Question 7

Evaluate $$\displaystyle \$$int_0$$^1 3x^2$\,dx.$$

Accepted Answer

$$1$$

Question 8

Let $$f(x) = x^4$ - 4x^3$. Find all critical points and classify them as local maxima, minima, or neither.

Accepted Answer

Critical points at $$x = 0 ($$local max), $$x = 3 ($$local min)

Question 9

Which of the following is the correct application of the Second Fundamental Theorem of Calculus for $$F(x) = \$$int_2$$^x \sqrt{t^2$ + 1}\,dt$$?

Accepted Answer

$$F'(x) = \sqrt{x^2$ + 1}$$

Question 10

A sand pile forms a cone whose height is always twice the radius of the base. If the radius increases at $$0.75 in/$$min, how fast is the volume increasing when the radius is $$6 in$$? (Leave your answer in terms of $$\pi.)$$

Accepted Answer

$$54\pi in$^3/$$min

Question 11

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{5x^2$ + 3}{2x^2$ - x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 12

Which of the following is NOT an indeterminate form for limits?

Accepted Answer

$$\frac{1}{0}$$

Question 13

Find $$\frac{dy}{dx} if$$ $$x^2$$ + $$y^2 = 25$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = -\frac{x}{y}$$

Question 14

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$Which theorem guarantees there exists $$c \in (a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$?

Accepted Answer

Mean Value Theorem

Question 15

Evaluate $$\displaystyle \int x e$$^{x^2}$$ dx.$$

Accepted Answer

$$\frac{1}{2} e$$^{x^2}$$ + C$$

Question 16

A company wants to minimize the cost of running a cable from point A (9 miles east of C) to point B (6 miles north of C), with land cost $$400/mi$$ and underwater cost $$500/mi. $$Which equation should be minimized to find the optimal distance $$x ($$miles) from C along the shore where the cable enters the water?

Accepted Answer

$$400(9 - x) + 500\sqrt{x^2$ + 36}$$

Question 17

Which of the following best describes the relationship between the sign of $$f''(x)$$ and the concavity of $$f(x)$$?

Accepted Answer

If $$f''(x) \u003e 0$$, $$f(x) is $$concave up; if $$f''(x) \u003c 0$$, $$f(x) is $$concave down.

Question 18

Find all $$x-$$values where $$f(x) = |x| is $$not differentiable.

Accepted Answer

$$x = 0$$

Question 19

Evaluate $$\displaystyle \$$int_0$$^\pi \sin x\,dx.$$

Accepted Answer

$$2$$

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