Q: Is the function $$f(x)$$ continuous at $$x=0$$ based on the graph?

No, because $$\lim_{x \to 0} f(x) \neq f(0)$$

Q: Find the derivative of $$f(x) = \sec^4$(9x-10).$$

$$36\sec^4$(9x-10)\tan(9x-10)$$

Question 1

Given the graph of $$f(x)$$, what is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$3$$

Question 2

Is the function $$f(x)$$ continuous at $$x=0$$ based on the graph?

Accepted Answer

No, because $$\lim_{x 	o 0} f(x) 
eq f(0)$$

Question 3

For the function $$g(x)$$ shown in the second graph, what is $$g(-2)$$?

Accepted Answer

$$5$$

Question 4

Evaluate $$\lim_{x 	o 2} g(x)$$ using the graph of $$g(x).$$

Accepted Answer

$$-2$$

Question 5

Is $$g(x)$$ continuous at $$x=0$$?

Accepted Answer

No, because there is a jump discontinuity at $$x=0$$

Question 6

Find $$\lim_{x 	o 3} \frac{x^2$-9}{x^2$-4x+1}.$$

Accepted Answer

$$-1$$

Question 7

Determine if $$f(x) = x+5 is $$continuous at $$x = -5.$$

Accepted Answer

Yes, because $$f(x) is a $$polynomial and polynomials are continuous everywhere.

Question 8

Where is the function $$f(x) = \begin{cases} 5 & x \u003c 0 \ 2x + x - 5 & 0 \leq x \u003c 2 \ x + 3 & x \geq 2 \end{cases}$$ discontinuous?

Accepted Answer

At $$x = 0$$ and $$x = 2$$

Question 9

Find the derivative of $$f(x) = 2x^6$ + x^5$ + 9.$$

Accepted Answer

$$12x^5$$ + $$5x^4$$

Question 10

Find $$\frac{dy}{dx}$$ for $$y = (4x$$^{-3}$$ + x) \cos x.$$

Accepted Answer

$$(-12x^{-4} + 1$$)\cos x + $$(4x^{-3} + x$$)(-\sin x)$$

Question 11

Find the derivative of $$g(x) = \sqrt{x}.$$

Accepted Answer

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

Question 12

Find the derivative of $$y = \sin^2$(x^5$$)$.

Accepted Answer

$$10x^4$$ \$$sin(x^5$$) \$$cos(x^5$$)$$

Question 13

A forest fire spreads in a circle with radius changing at a rate of $$3$$ feet per minute. When the radius is $$420$$ feet, at what rate is the area increasing? (Use $$A = \pi r^2$)

Accepted Answer

$$2520\pi$$ square feet per minute

Question 14

A cop is parked $$45$$ feet from a highway and measures the distance to a car passing a road sign $$60$$ feet down the highway. If the distance from the cop to the car is increasing at $$88 ft/sec$$, what is the speed of the car when it passes the sign?

Accepted Answer

$$60\sqrt{3} ft/sec$$

Question 15

Find the derivative of $$f(x) = \sec^4$(9x-10).$$

Accepted Answer

$$36\sec^4$(9x-10)	an(9x-10)$$

Question 16

Find the derivative of $$G(x) = (-3x^4$ - x^3$ + 100)^8$.

Accepted Answer

$$8(-3x^4$$ - $$x^3$$ + $$100)^7(-12x^3$ - 3x^2$)$

Question 17

Find the derivative of $$F(x) = \frac{8x^3$}{(2x-5)(x^2$ + x + 1)}$$ using the quotient rule.

Accepted Answer

$$\frac{24x^2$(2x-5)(x^2+x+1$$) - $$8x^3$$[$$2(x^2+x+1) + (2x-5)(2x+1$$)]}{[$$(2x-5)(x^2+x+1$$)]^2}$

MATH 165 Calculus Midterm Review – Step-by-Step Guidance

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