Q: Find the derivative of $$y = 5x \sec x \tan x$$ with respect to $$x.$$

$$5 \sec x \tan x + 5x \sec x \sec x$$

Question 1

Evaluate the following limit: $$\lim\limits_{x 	o -1} \frac{2x^2$ + 7x + 12}{x + 4}$$

Accepted Answer

$$-1$$

Question 2

A projectile is fired vertically upward into the air, and its position in meters above the ground after $$t$$ seconds is given by $$s(t) = -16t^2$ + 92t. $$What is the velocity of the projectile at $$t = 3$$ seconds?

Accepted Answer

$$44 m/s$$

Question 3

Find the derivative of $$y = 5x \sec x 	an x$$ with respect to $$x.$$

Accepted Answer

$$5 \sec x 	an x + 5x \sec x \sec x$$

Question 4

Determine the limit at infinity: $$\lim\limits_{x 	o \infty} \frac{-2x^2$ + 7x - 17}{4x^2$ + 5x + 16}$$

Accepted Answer

$$-\frac{1}{2}$$

Question 5

Find the equation of the tangent to the curve $$y = 6x - 11 at$$ $$x = 2.$$

Accepted Answer

$$y = 6x - 7$$

Question 6

A circle's radius is increasing at a constant rate of $$0.2 m/s. At $$what rate does the area of the circle increase when the radius is $$10 m$$? (Area $$A = \pi r^2$)

Accepted Answer

$$4\pi m$^2/s$$

Question 7

Find the derivative of $$y = \sin$$^{-1}$$ \sqrt{x}$$ with respect to $$x.$$

Accepted Answer

$$\frac{1}{2\sqrt{x} \sqrt{1-x}}$$

Question 8

Given $$y = x^2$ \arccos 2x$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$2x \arccos 2x - \frac{2x^2$}{\sqrt{1-4x^2$}}$$

Question 9

Find the vertical asymptotes of the function $$f(x) = \frac{2x^2$ - 7x + 12}{x^2$ - 5x + 6}.$$

Accepted Answer

$$x = 2$$ and $$x = 3$$

Question 10

A medication is administered and its concentration in the bloodstream at time $$t (in $$hours) is given by $$C(t) = 5e$$^{-0.5t}$$. What is the rate of change of the concentration at t = 2$ hours?

Accepted Answer

$$-1.84$$

Question 11

Find the derivative of $$y = \ln(6x)$$ with respect to $$x.$$

Accepted Answer

$$\frac{1}{x}$$

Question 12

Evaluate the following limit: $$\lim\limits_{x 	o \infty} \frac{3x^2$ - 5x + 4}{5x^2$ + 6x - 2}$$

Accepted Answer

$$\frac{3}{5}$$

Question 13

Given $$y = x^2$ + 5x^4$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2x + 20x^3$

Question 14

A bacteria population grows according to $$P(t) = 100e$$^{0.03t}$$, where t$ is in hours. What is the instantaneous rate of growth at t = 10$ hours?

Accepted Answer

$$100e^{0.3}$$ 	imes 0.03$$

Question 15

Find the derivative of $$y = e$$^{x^2}$$ with respect to x$.

Accepted Answer

$$2x e$$^{x^2}$$

Question 16

If $$y = 	an(2x^2$ + 5)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$4x \sec^2$(2x^2 + 5$$)$

Question 17

Find the derivative of $$y = \frac{3x^2$ + 5}{x^2$ - 1}$$ with respect to $$x.$$

Accepted Answer

$$\frac{6x(x^2$ - 1) - 2x(3x^2$ + 5)}{(x^2$ - 1)^2$}$$

Question 18

A car's position is given by $$s(t) = 2t^3$ - 7t^2$ + 10t + 4. $$What is the car's acceleration at $$t = 2$$ seconds?

Accepted Answer

$$-2$$

Question 19

Find $$\frac{dy}{dx} if$$ $$y = x^2$ \sqrt{x + 5}.$$

Accepted Answer

$$x(x + 10) / \sqrt{x + 5}$$

Question 20

Use implicit differentiation to find $$\frac{dy}{dx} if$$ $$2xy - y^2$ = 1.$$

Accepted Answer

$$\frac{y}{x - y}$$

Calculus I: Limits, Derivatives, and Applications – Study Guide

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