Q: If $$y = \sec x \cdot \csc x$$, what is $$\frac{dy}{dx}$$?

$$\sec x \csc x (\tan x - \cot x)$$

Question 1

A rock is launched vertically upward from the ground with an initial speed of 96 ft/s. Its position after $$t$$ seconds is given by $$s(t) = -16t^2$ + 96t. $$What is the instantaneous velocity of the rock at $$t = 1$$ second?

Accepted Answer

$$64 ft/s$$

Question 2

Given $$f(x) = x^2$ + 4x$$, what is $$f'(x)$$?

Accepted Answer

$$2x + 4$$

Question 3

Suppose the speed of a car approaching a stop sign is given by $$v(t) = t^3$ - 5t^2$ for 0$$ \leq t \leq 5$$. What is the car's acceleration at t = 1$?

Accepted Answer

$$a(1) = 3(1)^2$ - 10(1) = -7 m/s$^2$$

Question 4

Which of the following statements is true about differentiability and continuity?

Accepted Answer

If a function is differentiable at $$a$$, then it is also continuous at $$a.$$

Question 5

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

Accepted Answer

$$4e^x - 6x$$

Question 6

Let $$f(x) = \sqrt{x}. $$What is the equation of the tangent line to $$f at $$the point $$(4,2)$$?

Accepted Answer

$$y = \frac{1}{4}(x - 4) + 2$$

Question 7

If $$y = \sin x^2$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2x \cos x^2$

Question 8

Suppose the cost function for producing $$x$$ items is $$C(x) = -0.02x^2$ + 50x + 100. $$What is the marginal cost function?

Accepted Answer

$$C'(x) = -0.04x + 50$$

Question 9

Find the derivative of $$y = 	an x + 10^3$.

Accepted Answer

$$\sec^2$ x$$

Question 10

Given $$f(x) = x^3$ - 15x^2$ + 24x$$, for what value(s) of $$x$$ does the tangent line to the graph of $$f$$ have a slope of $$6$$?

Accepted Answer

$$x = 1 or$$ $$x = 4$$

Question 11

If $$y = \sec x \cdot \csc x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\sec x \csc x (	an x - \cot x)$$

Question 12

Find the derivative of $$f(x) = (5x + 4)^3$ using the chain rule.

Accepted Answer

$$15(5x + 4)^2$

Question 13

Suppose $$y = \sin(\cos x). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$-\cos(\cos x) \sin x$$

Question 14

If $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 15

The demand for processed pork in Canada is described by $$D(p) = 286 - 20p. $$What is the price elasticity of demand at $$p = 10$$?

Accepted Answer

$$E(10) = -\frac{200}{86}$$

Question 16

Find $$\frac{d}{dx} \left( \frac{3x^2$}{2x^3$ + 4} \right)$$ using the quotient rule.

Accepted Answer

$$\frac{6x(2x^3$ + 4) - 3x^2$(6x^2$$)}{$$(2x^3$$ + $$4)^2$$}$

Question 17

If $$y = \sin x$$, what is $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

$$-\sin x$$

Question 18

Suppose $$y = x^2$ + xy - y^2$ = 7$$ and the point $$(3,2) is on $$the curve. What is the slope of the tangent line at this point?

Accepted Answer

$$\frac{1}{5}$$

Question 19

Which of the following is NOT a reason a function is not differentiable at a point?

Accepted Answer

It is continuous and smooth at the point.

Calculus I: Derivatives, Tangents, and Rates of Change – Comprehensive Study Notes

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