$$\lim_{h \to 0} \frac{12h + 3h^2$ + 6hx}{h}$$

Question 1

Suppose $$p(x) = 4x^2$ + 2x^4$ + \frac{4}{x^3$} + 4. $$What is the equation of the tangent line to $$p(x) at$$ $$x = 1$$?

Accepted Answer

$$y = 4x + 10$$

Question 2

Find the derivative of $$q(x) = \frac{3x^3$ + 9}{5x^2$ + 6}$$ using the quotient rule.

Accepted Answer

$$\frac{(5x^2$ + 6)(9x^2$) - (3x^3$ + 9)(10x)}{(5x^2$ + 6)^2$}$$

Question 3

An object is thrown straight up. Its height after $$t$$ seconds is $$h(t) = -9t^2$ + 75t^2$ + 173. At $$what time does it reach its maximum height?

Accepted Answer

$$t = 16.6667$$

Question 4

Suppose the profit of a product is modeled by $$f(x) = 2x^3$ - 30x^2$ + 126x + 9. $$What is the absolute maximum value of $$f(x) on$$ $$[4,8]$$?

Accepted Answer

$$f = 161$$

Question 5

Find the critical point of $$f(x) = 5e$$^{0.2x^2 - 3x}$$.

Accepted Answer

$$x = 7.5$$

Question 6

Evaluate $$\lim_{x 	o -2} \frac{\sqrt{x+27} - 5}{x+2}.$$

Accepted Answer

$$0.1$$

Question 7

Given $$4x^2 + 8xy$$ + $$2y^2 = 15$, find $$\frac{dy}{dx}$$ by implicit differentiation.

Accepted Answer

$$\frac{-12x^2$ + 8y}{8x + 4y}$$

Question 8

Suppose $$f(x, y) = \sqrt{x} + \sqrt{y}. $$What is the domain of $$f(x, y)$$?

Accepted Answer

The set of all points $$(x, y) in $$the $$xy-$$plane where $$x \geq 0$$ and $$y \geq 0$$

Question 9

Find the horizontal asymptote of $$f(x) = \frac{2x - 4}{4x^2$ + 5x - 2}.$$

Accepted Answer

$$y = 0$$

Question 10

If a differentiable function $$f(x)$$ satisfies $$f(-4) = f(4)$$, then by Rolle's Theorem, there exists $$c in$$ $$(-4, 4)$$ such that:

Accepted Answer

$$f'(c) = 0$$

Question 11

If $$f(x) = 3 + 2x^2$ + e$$^{5x}$$, what is f$$^{(3)}$$(x)$$?

Accepted Answer

$$125e^{5x}$$

Question 12

Let $$f(t) = -2t^3$ + 27t^2$ - 108t + 120. On $$which interval is $$f(t)$$ increasing?

Accepted Answer

$$(3, 6)$$

Question 13

Evaluate $$\lim_{x 	o 7} \frac{2x^3$ - 18x^2$ + 28x}{x^2$ - 11x + 28}$$ rounded to 4 decimal places.

Accepted Answer

$$23.3333$$

Question 14

Evaluate $$\lim_{x 	o \infty} \frac{2x^3$ + 6x^2$ + 3x - 6}{2x^2$ - 4x + 15}.$$

Accepted Answer

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

Question 15

Let $$f(x) = 3(x+2)^2$. Using the definition of the derivative, f$$'(x)$$ is equal to:

Accepted Answer

$$\lim_{h 	o 0} \frac{12h + 3h^2$ + 6hx}{h}$$

Question 16

Find the derivative of $$h(x) = \frac{e$$^{-3}$$}{3 + e$$^{(2x^2 + 10)}$$}.$$

Accepted Answer

$$\frac{-e$$^{-3}$$(2x^2$$ + $$10)(4x)e^{(2x^2 + 10)}$$}{(3 + $$e^{(2x^2 + 10)})^2$}$$

Question 17

The domain of $$g(x) = \frac{88 \sqrt{x-5}}{15x^2$ - 75x + 90}$$ can be written as:

Accepted Answer

$$[-2, 3) \cup (3, \infty)$$

Step-by-Step Calculus Study Guidance for Practice Final Exam

Study Guide - Practice Questions