Q: A medical device records $$y = 5\tan x - \sec x + 4\tan x. $$What is $$\frac{dy}{dx}$$?

$$y' = 5\sec^2$ x - \sec x \tan x + 4\sec^2$ x$$

Question 1

A pharmaceutical company models the concentration of a drug in the bloodstream by $$f(x) = 3x^2$ - 5x + 2. $$What is the slope of the tangent line to the concentration curve at $$x = 2$$ hours?

Accepted Answer

$$7$$

Question 2

Using the limit definition of the derivative, find $$f'(x)$$ for $$f(x) = x^2$.

Accepted Answer

$$2x$$

Question 3

A car's position is given by $$f(x) = 4x^3$ - 3x^2$ + 5. $$What is the instantaneous velocity at $$x = 1$$?

Accepted Answer

$$8$$

Question 4

Find the equation of the tangent line to $$f(x) = x^3$ - 4 at $$the point $$(2, 4).$$

Accepted Answer

$$y = 12x - 20$$

Question 5

A function $$f(x) = x^2$ + 4$$ has a tangent line parallel to $$2x - 3y + 2 = 0. $$What is the equation of this tangent line?

Accepted Answer

$$y = \frac{2}{3}x + \frac{9}{2}$$

Question 6

For $$f(x) = x^3$ + 2x$$, use the alternate form of the derivative to find $$f'(1).$$

Accepted Answer

$$5$$

Question 7

Which $$x-$$values is $$f(x) = \frac{3x^2$ - 2x}{x^2$ - 5}$$ differentiable?

Accepted Answer

All real values of $$x$$, except $$x = 1$$ and $$x = 2$$

Question 8

Find the derivative of $$y = 4x^3$ + 15x - 7.$$

Accepted Answer

$$y' = 12x^2$ + 15$$

Question 9

A medical device records $$y = 5	an x - \sec x + 4	an x. $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$y' = 5\sec^2$ x - \sec x 	an x + 4\sec^2$ x$$

Question 10

Find the slope of the tangent line to $$y = 3x^2$ - 5 at$$ $$x = 2.$$

Accepted Answer

$$12$$

Question 11

A ladder 20 feet long is leaning against a wall. The base is pulled away at $$2 ft/sec. $$How fast is the top moving down the wall when the base is $$10 ft $$from the wall?

Accepted Answer

$$\frac{10\sqrt{3}}{3} ft/sec$$

Question 12

The radius $$r of a $$circle increases at $$5 cm/$$min. What is the rate of change of the area when $$r = 12 cm$$?

Accepted Answer

$$120\pi cm$^2/$$min

Question 13

All edges of a cube expand at $$4 cm/sec. $$How fast is the volume changing when each edge is $$6 cm$$?

Accepted Answer

$$432 cm$^3/sec$$

Question 14

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

Accepted Answer

$$f'(x) = 20x^3$ - 6x + 4$$

Question 15

Find the second derivative of $$f(x) = 5\cos 3x.$$

Accepted Answer

$$f''(x) = -45\cos 3x$$

Question 16

A function $$f(x) = (x^2$ - 4)(x + 1)$$ has a horizontal tangent line at which point?

Accepted Answer

$$(2, -3)$$

Question 17

Use the product rule to differentiate $$f(x) = x^3$ \sin x.$$

Accepted Answer

$$f'(x) = 3x^2$ \sin x + x^3$ \cos x$$

Question 18

Use the quotient rule to differentiate $$f(x) = \frac{x^4$}{2x - 2}.$$

Accepted Answer

$$f'(x) = \frac{4x^3$(2x - 2) - x^4$ \cdot 2}{(2x - 2)^2$}$$

Question 19

Find $$\frac{dy}{dx} by $$implicit differentiation for $$x^2 + 4x y$$ + $$3y^2 = 49$.

Accepted Answer

$$\frac{dy}{dx} = \frac{2x + 4y}{4x + 6y}$$

Step-by-Step Calculus Practice Test Guidance (Derivatives, Tangents, Related Rates, and More)

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