Calculus Maximus
WS 4.1: Tangent Line Problem
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Worksheet 4.1—Tangent Line Problem
Show all work.
1. Find the slope of the tangent lines to the graphs of the following functions at the indicated points. Use
the alternate form.
(a) f ( x ) = 3 − 2 x at ( −1,5)
(b) g ( x ) = 5 − x 2 at x = 2
dy
, of each of the following using the limit definition.
dx
3
(b) y =
(c) f ( x ) = x − 2
x −1
2. Find the derivative function, either f ʹ′ ( x ) or
(a) f ( x ) = 2 x 2 + 3x − 4
Calculus Maximus
WS 4.1: Tangent Line Problem
3. For each of the following, find the equation of BOTH the tangent line and the normal line to the
function at the indicated points. Use either limit definition of the derivative.
(a) g ( x ) = x 2 + 1 at ( 2,5)
(b) y = x − 1 at x = 9
4. Find an equation of the line that is tangent to f ( x ) = x3 and parallel to the line 3x − y + 1 = 0
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Calculus Maximus
WS 4.1: Tangent Line Problem
5. For each of the following functions f ( x ) , on the same coordinate grid, sketch a graph of f ʹ′ ( x ) .
(a)
(b)
(c)
(d)
(e)
(g)
6. Sketch a function that has the following characteristics.
f ( 0) = 4 , f ʹ′ ( 0 ) = 0 , f ʹ′ ( x ) < 0 for x < 0 , and f ʹ′ ( x ) > 0 for x > 0
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(f)
Calculus Maximus
7. If f ʹ′ ( c ) = 3, find f ʹ′ ( −c ) if
(a) f is an odd function, and if
WS 4.1: Tangent Line Problem
(b) f is an even function.
8. Find the equations of the two tangent lines to f ( x ) = x 2 that pass through the point (1, −3) , a point not
on the graph of f. Hint: Sketch the scenario first.
9. For each of the following, the limit represents f ʹ′ ( c ) for a function f ( x ) and a number x = c . Find
both f and c.
3
⎡⎣5 − 3 (1 + h )⎤⎦ − 2
−2 + h ) + 8
(
(a) lim
(b) lim
h →0
h→0
h
h
− x 2 + 36
x →6
x−6
(c) lim
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2 x −6
x →9 x − 9
(d) lim
Calculus Maximus
WS 4.1: Tangent Line Problem
10. Use alternate form to discuss the differentiability of each of the following at the point.
⎧⎪( x − 1)3 , x ≤ 1
3
(a) f ( x ) = x + 2 x at x = 1
(b) f ( x ) = ⎨
at x = 1
2
⎪⎩( x − 1) , x > 1
⎧ x 2 + 1, x ≤ 2
at x = 2
(c) f ( x ) = ⎨
4
x
−
3,
x
>
2
⎩
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⎧ 1
⎪ x + 1, x < 2
(d) f ( x ) = ⎨ 2
at x = 2
⎪ 2 x , x ≥ 2
⎩
Calculus Maximus
WS 4.1: Tangent Line Problem
11. True or False. If false, explain why or give a counterexample.
(a) The slope of the tangent line to the differentiable function f at the point ( 2, f ( 2 )) is
f ( 2 + h ) − f ( 2)
.
h
(b) If a function is continuous at a point, then it is differentiable at that point.
(c) If a function has derivatives from both the right and the left at a point, then it is differentiable at that
point.
(d) If a function is differentiable at a point, then it is continuous at that point.
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