Question 1

Given the function $$y = -5 - 5x^2$, find the slope of the curve at the point P(3$$, -50)$$.

Accepted Answer

$$-30$$

Question 2

For the function $$y = -5 - 5x^2$, what is the equation of the tangent line at P(3$$, -50)$$?

Accepted Answer

$$y = -30x + 40$$

Question 3

Evaluate the limit: $$\lim_{x 	o 5} \frac{-x^2$ + 8x - 1}{2}.$$

Accepted Answer

$$6$$

Question 4

Find the limit: $$\lim_{x 	o -1} \frac{x+1}{x^2$ + 7x + 6}.$$

Accepted Answer

$$0$$

Question 5

Find the limit: $$\lim_{x 	o 5} \frac{x^2$ - 2x - 15}{x - 5}.$$

Accepted Answer

$$8$$

Question 6

Find the limit: $$\lim_{x 	o 1} \frac{x - x^2$}{1 - x}.$$

Accepted Answer

$$2$$

Question 7

Find the limit: $$\lim_{x 	o 3} \frac{x^2$ + 40 - 7}{x - 3}.$$

Accepted Answer

$$23$$

Question 8

Evaluate the right-hand and left-hand limits for $$f(x) = |x + 9| + 11 at$$ $$x = -9. $$What are $$\lim_{x 	o -9^+} f(x)$$ and $$\lim_{x 	o -9^-} f(x)$$?

Accepted Answer

$$11$$ and $$-7$$

Question 9

Using $$\lim_{	heta 	o 0} \frac{\sin 	heta}{	heta} = 1$$, evaluate $$\lim_{x 	o 0} x \csc 8x \cos 2x.$$

Accepted Answer

$$0$$

Question 10

For what values of $$a$$ and $$b is $$the function $$f(x) = \begin{cases} -2 & x \leq -1 \ ax - b & -1 \u003c x \u003c 1 \ 5 & x \geq 1 \end{cases}$$ continuous at every $$x$$?

Accepted Answer

$$a = 3.5, b = 1.5$$

Question 11

Find the limit as $$x 	o \infty$$ and $$x 	o -\infty$$ for $$h(x) = \frac{-3x^3$ - 4x^2$ + 3}{5x^3$ + 4x^2$ - 2x}.$$

Accepted Answer

$$-\frac{3}{5}$$ and $$-\frac{3}{5}$$

Question 12

Find the first and second derivatives of $$r = -\frac{1}{2s^6$} + 5s^4$ with respect to s$.

Accepted Answer

$$\frac{3}{s^7$} + 20s^3$ and $$-\frac{21}{$$s^8$$} + $$60s^2$$

Question 13

Find the derivative of $$s = (4 - t)^3$ + t^2$ - 1$$ with respect to $$t.$$

Accepted Answer

$$-3(4 - t)^2$ + 2t$$

Question 14

Find the derivative of $$r = 5e^w + 9w$$ with respect to $$w.$$

Accepted Answer

$$5e^w + 9$$

Question 15

Find all points $$(x, y) on $$the graph $$y = x^2$ - x$$ with tangent lines perpendicular to the line $$y = x - 4.$$

Accepted Answer

$$(0, 0)$$ and $$(2, 2)$$

Question 16

Find the value of $$a$$ that makes the function $$g(x) = \begin{cases} ax & x \u003c 0 \ x^2$ - 7x & x \geq 0 \end{cases}$$ differentiable for all values of $$x.$$

Accepted Answer

$$-7$$

Question 17

A body moves according to $$s = -t^3$ + 15t^2$ - 75t$$ for $$0 \leq t \leq 5. $$What is the displacement over this interval?

Accepted Answer

$$0$$

Question 18

Find $$\frac{dy}{dx}$$ for $$y = 10x + 7\cos x.$$

Accepted Answer

$$10 - 7\sin x$$

Question 19

Find $$\frac{dy}{dx}$$ for $$y = x^6$ \cos x.$$

Accepted Answer

$$6x^5$$ \cos x - $$x^6$$ \sin x$$

Calculus I Test 1 Review – Step-by-Step Guidance

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